Recent zbMATH articles in MSC 52https://zbmath.org/atom/cc/522022-11-17T18:59:28.764376ZWerkzeugIslamic geometric patterns in higher dimensionshttps://zbmath.org/1496.000532022-11-17T18:59:28.764376Z"Moradzadeh, Sam"https://zbmath.org/authors/?q=ai:moradzadeh.sam"Nejad Ebrahimi, Ahad"https://zbmath.org/authors/?q=ai:ebrahimi.ahad-nejadSummary: The purpose of this paper is to develop the Islamic geometric patterns from planar coordinates to three or higher dimensions through their repeat units. We use historical plane methods, polygons in contact (PIC) and point-joined, in our deductive approaches. The mentioned approach makes use of a novel method of tessellation that generates 3D Islamic patterns called ``interior polyhedral stellations''. The outputs showed that both the PIC and point-joined methods have strengths and weaknesses. Point-joined stellations are more efficient for regular repeat units and PIC is suitable for complex designs. These two methods can produce a large range of patterns and can be employed simultaneously. This study effectively answers the question regarding the gap between planar design from Muslim achievements and contemporary demands in modern art and architecture. We also propose techniques for constructing aperiodic three-dimension Islamic geometric patterns tessellation and two-point family.Kazhdan-Lusztig polynomials of fan matroids, wheel matroids, and whirl matroidshttps://zbmath.org/1496.050212022-11-17T18:59:28.764376Z"Lu, Linyuan"https://zbmath.org/authors/?q=ai:lu.linyuan"Xie, Matthew H. Y."https://zbmath.org/authors/?q=ai:xie.matthew-h-y"Yang, Arthur L. B."https://zbmath.org/authors/?q=ai:yang.arthur-li-bo|yang.arthur-l-bSummary: The Kazhdan-Lusztig polynomial of a matroid was introduced by \textit{B. Elias} et al. [Adv. Math. 299, 36--70 (2016; Zbl 1341.05250)]. The properties of these polynomials need to be further explored. In this paper we prove that the Kazhdan-Lusztig polynomials of fan matroids coincide with Motzkin polynomials, which was conjectured by \textit{K. R. Gedeon} [Electron. J. Comb. 24, No. 3, Research Paper P3.12, 10 p. (2017; Zbl 1369.05029)]. As a byproduct, we determine the Kazhdan-Lusztig polynomials of graphic matroids of squares of paths. We further obtain explicit formulas of the Kazhdan-Lusztig polynomials of wheel matroids and whirl matroids. We prove the real-rootedness of the Kazhdan-Lusztig polynomials of these matroids, thus providing positive evidence for a conjecture due to \textit{K. Gedeon} et al. [J. Comb. Theory, Ser. A 150, 267--294 (2017; Zbl 1362.05131)]. Based on the results on the Kazhdan-Lusztig polynomials, we also determine the \(Z\)-polynomials of fan matroids, wheel matroids and whirl matroids, and prove their real-rootedness, thus providing further evidence in support of a conjecture of \textit{N. Proudfoot} et al. [Electron. J. Comb. 25, No. 1, Research Paper P1.26, 21 p. (2018; Zbl 1380.05022)].The facets of the matroid polytope and the independent set polytope of a positroidhttps://zbmath.org/1496.050222022-11-17T18:59:28.764376Z"Oh, SuHo"https://zbmath.org/authors/?q=ai:oh.suho"Xiang, David"https://zbmath.org/authors/?q=ai:xiang.davidSummary: A positroid is a special case of a realizable matroid that arose from the study of the totally nonnegative part of the Grassmannian by \textit{A. Postnikov} [``Total positivity, Grassmannians, and networks'', Preprint, \url{arXiv:math/0609764}]. In this paper, we study the facets of its matroid polytope and the independent set polytope. This allows one to describe the bases and independent sets directly from the decorated permutation, bypassing the use of the Grassmann necklace. We also describe a criterion for determining whether a given cyclic interval is a flat or not using the decorated permutation, then show how it applies to checking the concordancy of positroids.Unbreakable matroidshttps://zbmath.org/1496.050232022-11-17T18:59:28.764376Z"Oxley, James"https://zbmath.org/authors/?q=ai:oxley.james-g"Pfeil, Simon"https://zbmath.org/authors/?q=ai:pfeil.simonSummary: A matroid \(M\) is unbreakable if \(M\) is connected and, for each flat \(F\), the matroid \(M / F\) is connected. Equivalently, \(M\) is unbreakable if its dual has no two skew circuits. This paper characterizes unbreakable matroids in terms of excluded parallel minors and determines all regular unbreakable matroids.Blossoming bijection for bipartite pointed maps and parametric rationality of general maps of any surfacehttps://zbmath.org/1496.050782022-11-17T18:59:28.764376Z"Dołęga, Maciej"https://zbmath.org/authors/?q=ai:dolega.maciej"Lepoutre, Mathias"https://zbmath.org/authors/?q=ai:lepoutre.mathiasSummary: We construct an explicit bijection between bipartite pointed maps of an arbitrary surface \(\mathbb{S} \), and specific unicellular blossoming maps of the same surface. Our bijection gives access to the degrees of all the faces, and distances from the pointed vertex in the initial map. The main construction generalizes recent work of the second author which covered the case of an orientable surface. Our bijection gives rise to a first combinatorial proof of a parametric rationality result concerning the bivariate generating series of maps of a given surface with respect to their numbers of faces and vertices. In particular, it provides a combinatorial explanation of the structural difference between the aforementioned bivariate parametric generating series in the case of orientable and non-orientable maps.The convex and weak convex domination number of convex polytopeshttps://zbmath.org/1496.051312022-11-17T18:59:28.764376Z"Maksimović, Zoran Lj."https://zbmath.org/authors/?q=ai:maksimovic.zoran-lj"Savić, Aleksandar Lj."https://zbmath.org/authors/?q=ai:savic.aleksandar-lj"Bogdanović, Milena S."https://zbmath.org/authors/?q=ai:bogdanovic.milena-sSummary: This paper is devoted to solving the weakly convex dominating set problem and the convex dominating set problem for some classes of planar graphs -- convex polytopes. We consider all classes of convex polytopes known from the literature and present exact values of weakly convex and convex domination number for all classes, namely \(A_n\), \(B_n\), \(C_n\), \(D_n\), \(E_n\), \(R_n\), \(R^{\prime\prime}_n\), \(Q_n\), \(S_n\), \(S^{\prime\prime}_n\), \(T_n, T^{\prime\prime}_n\) and \(U_n\). When \(n\) is up to 26, the values are confirmed by using the exact method, while for greater values of \(n\) theoretical proofs are given.Independent resolving number of convex polytopeshttps://zbmath.org/1496.051372022-11-17T18:59:28.764376Z"Suganya, B."https://zbmath.org/authors/?q=ai:suganya.b"Arumugam, S."https://zbmath.org/authors/?q=ai:arumugam.senthil-m|arumugam.subramanian|arumugam.saravanan|arumugam.sivabalanSummary: Let \(G=(V,E)\) be a connected graph. Let \(W=\{w_1,w_2,\dots,w_k\}\) be a subset of \(V\) with an order imposed on it. Then \(W\) is called a resolving set for \(G\) if for every two distinct vertices \(x,y\in V(G)\), there is a vertex \(w_i\in W\) such that \(d(x,w_i)\neq d(y,w_i)\). The minimum cardinality of a resolving set of \(G\) is called the metric dimension of \(G\) and is denoted by \(\dim(G)\). A subset \(W\) is called an independent resolving set for \(G\) if \(W\) is both independent and resolving. The minimum cardinality of an independent resolving set in \(G\) is called the independent resolving number of \(G\) and is denoted by \(\operatorname{ir}(G)\). In this paper we determine the independent resolving number \(\operatorname{ir}(G)\) for three classes of convex polytopes.
For the entire collection see [Zbl 1369.68008].What convex geometries tell about shattering-extremal systemshttps://zbmath.org/1496.051812022-11-17T18:59:28.764376Z"Chornomaz, Bogdan"https://zbmath.org/authors/?q=ai:chornomaz.bogdanSummary: We give a characterization of shattering-extremal set systems in terms of forbidden projections, in the spirit of Dietrich's characterization of antimatroids. Apart from that, we prove several metric and topological properties of such systems, which, however, do not amount to a characterization. The ideas for all these results come from the similar characterizations of antimatroids and convex geometries, and due to the fact that both of them are special cases of shattering-extremal systems.New interpretations of the higher Stasheff-Tamari ordershttps://zbmath.org/1496.051872022-11-17T18:59:28.764376Z"Williams, Nicholas J."https://zbmath.org/authors/?q=ai:williams.nicholas-jSummary: \textit{P. H. Edelman} and \textit{V. Reiner} [Mathematika 43, No. 1, 127--154 (1996; Zbl 0854.06003)] defined the two higher Stasheff-Tamari orders on triangulations of cyclic polytopes and conjectured them to coincide. We open up an algebraic angle for approaching this conjecture by showing how these orders arise naturally in the representation theory of the higher Auslander algebras of type \(A\), denoted \(A_n^d\). For this we give new combinatorial interpretations of the orders, making them comparable. We then translate these combinatorial interpretations into the algebraic framework. We also show how triangulations of odd-dimensional cyclic polytopes arise in the representation theory of \(A_n^d\), namely as equivalence classes of \(d\)-maximal green sequences. We furthermore give the odd-dimensional counterpart to the known description of \(2d\)-dimensional triangulations as sets of non-intersecting \(d\)-simplices of a maximal size. This consists in a definition of two new properties which imply that a set of \(d\)-simplices produces a \((2 d + 1)\)-dimensional triangulation.Rotary one-facet maniplexeshttps://zbmath.org/1496.051912022-11-17T18:59:28.764376Z"Pellicer, Daniel"https://zbmath.org/authors/?q=ai:pellicer.daniel"Wilson, Steve"https://zbmath.org/authors/?q=ai:wilson.stephen-eSummary: Maniplexes are combinatorial objects that generalize, simultaneously, maps on surfaces and abstract polytopes. We are interested on studying rotary maniplexes, that is, those having maximal 'rotational' symmetry.
This note classifies rotary 4-dimensional maniplexes with the property of having exactly one facet, and gives examples and related results.On the Jacobian ideal of an almost generic hyperplane arrangementhttps://zbmath.org/1496.130052022-11-17T18:59:28.764376Z"Burity, Ricardo"https://zbmath.org/authors/?q=ai:burity.ricardo"Simis, Aron"https://zbmath.org/authors/?q=ai:simis.aron"Tohǎneanu, Ştefan O."https://zbmath.org/authors/?q=ai:tohaneanu.stefan-oLet \(\mathcal{A} = \{H_{1},\dots, H_{m}\}\) be a central hyperplane arrangement of rank \(n\) in \(\mathbb{K}^{n}\), where \(\mathbb{K}\) is a field of characteristic zero. For each \(i \in \{1,\dots, m\}\) let us denote by \(\ell_{i}\) a linear form such that \(\ker(\ell_{i})=H_{i}\), and consider \(f = f_{1} \cdots f_{m}\) the defining equation of \(\mathcal{A}\). Denote by \(J_{f}\) the Jacobian ideal associated with \(f\) generated by the partial derivatives of \(f\). The authors conjecture that \(J_{f}\) is a minimal reduction of the ideal \(\mathcal{I}\) which is generated by the \((m-1)\)-fold products of distinct forms among \(\ell_{1},\dots, \ell_{m}\). First of all, they verify this conjecture in the case when \(\mathcal{A}\) is almost generic, i.e., any \(n-1\) among the defining linear forms are linearly independent, and they confirm this conjecture for \(n=3\) unconditionally. Moreover, they show that in the case of \(n=3\) the ideal \(J_{f}\) is of linear type, i.e., the natural surjection from the symmetric algebra of \(J_{f}\) to its Rees algebra is an isomorphism. As a corollary, the authors show that if \(f \in \mathbb{K}[x,y,z]\) is the defining polynomial of a central hyperplane arrangement, then the Rees algebra of the Jacobian ideal \(J_{f}\) is Cohen-Macaulay.
Reviewer: Piotr Pokora (Kraków)Graded Cohen-Macaulay domains and lattice polytopes with short \(h\)-vectorhttps://zbmath.org/1496.130382022-11-17T18:59:28.764376Z"Katthän, Lukas"https://zbmath.org/authors/?q=ai:katthan.lukas"Yanagawa, Kohji"https://zbmath.org/authors/?q=ai:yanagawa.kohjiLet \(R=\bigoplus_{i\ge 0}R_i\) be a Noetherian graded Cohen-Macaulay domain with \(R_0=\Bbbk\) being an algebraically closed field. Let \(S=\Bbbk[R_1]\). Suppose that \(R\) is \textit{semi-standard graded}, namely, \(R\) is finitely generated as an \(S\)-module. The main result of this paper states that the graded Betti numbers \(\beta_{p,p+s}^S(R)=0\) for \(0\le p\le h_1-h_s\) where \((h_0,h_1,\dots,h_s)\) is the \(h\)-vector of \(R\). Green's vanishing theorem as presented by \textit{D. Eisenbud} and \textit{J. Koh} [Adv. Math. 90, No. 1, 47--76 (1991; Zbl 0754.13012)] is central in its proof. As an important corollary, if \(h_s\le h_1\), then \(R\) is generated by elements of degree \(\le s-1\) as an \(S\)-module.
This result is then applied to study the Ehrhart ring \(\Bbbk[P]\) of a lattice polytope \(P\subset \mathbb{R}^d\). Various bounds on different generators can be slightly improved. In particular, if the \(h\)-vector of \(\Bbbk[P]\) takes the form \((1,h_1^*,h_2^*)\) with \(h_2^*<h_1^*\), then \(P\) has \textit{integer decomposition property} (IDP).
Reviewer: Yi-Huang Shen (Hefei)Local numerical equivalences and Okounkov bodies in higher dimensionshttps://zbmath.org/1496.140072022-11-17T18:59:28.764376Z"Choi, Sung Rak"https://zbmath.org/authors/?q=ai:choi.sung-rak"Park, Jinhyung"https://zbmath.org/authors/?q=ai:park.jinhyung"Won, Joonyeong"https://zbmath.org/authors/?q=ai:won.joonyeongLet \(X\) be a smooth projective variety of dimension \(n\) and let \(D\) and \(D'\) be pseudoeffective divisors on \(X\). Recall that two divisors \(D\) and \(D'\) are numerically equivalent (\(D\equiv D'\)) if and only if \(D\cdot C=D'\cdot C\) for every irreducible curve \(C\) on \(X\). Let \(D=P+N\) be the divisorial Zariski decomposition. For a fixed point \(x\in X\), we can further decompose the negative part \(N=N_x+N_x^c\) into the effective divisors \(N_x\) and \(N_x^c\) such that every irreducible component of \(N_x\) passes through \(x\). We say that the decomposition
\[
D=P+N_x+N_x^c
\]
is the refined divisorial Zariski decomposition of \(D\) at a point \(x\).
We say that two pseudoeffective divisors \(D\) and \(D'\) are numerically equivalent near \(x\) (\(D\equiv_x D'\)) if \(P\equiv P'\) and \(N_x\equiv N_x'\). Now we decompose the divisor \(N_x\) as
\[
N_x=N_x^{\mathrm{sm}}+N_x^{\mathrm{sing}}
\]
where every irreducible component of \(N_x^{\mathrm{sm}}\) (resp. \(N_x^{\mathrm{sing}}\)) is smooth (resp. singular).
Let \(f: \widetilde{X}\rightarrow X\) be a birational morphism between smooth projective varieties of dimension \(n\) and let \(x\in X\) be a point. An admissible flag \(\widetilde{Y}_{\bullet}\) on \(\widetilde{X}\) is said to be
\begin{itemize}
\item centered at \(x\) if \(f(\widetilde{Y_n})=\{x\}\),
\item proper over \(X\) if \(\mathrm{codim} \, f(\widetilde{Y_i})=i\),
\item infinitesimal over \(X\) if \(\mathrm{codim} \, f(\widetilde{Y_n})=1\).
\end{itemize}
An admissible flag \(\widetilde{Y}_{\bullet}\) on \(\widetilde{X}\) that is proper over \(X\) is said to be induced (by an admissible flag \(Y_{\bullet}\) on \(X\)) if \(f(\widetilde{Y}_i)=Y_i\) for each \(0\leq i\leq n\).
An admissible flag \(\widetilde{Y}_{\bullet}\) on \(\widetilde{X}\) that is infinitesimal over \(X\) is said to be induced (by an admissible flag \(Y_{\bullet}\) on \(X\)) if there is a proper admissible flag \(Y_{\bullet}'\) on \(\widetilde{X}\) induced by \(Y_{\bullet}\) such that \(f(\widetilde{Y}_1)=Y_n\) and \(\widetilde{Y}_i=\widetilde{Y}_1\cap Y_{i-1}'\) for \(2\leq i\leq n\).
Let \(\Delta_{Y_{\bullet}}(D)\) be the Okounkov body of a divisor \(D\) with respect to an admissible flag \(Y_{\bullet}\).
The two main theorems of the paper give a generalization of Roe's results into higher dimensions.
{Theorem A.} Let \(D\) and \(D'\) be big divisors on a smooth projective varieties \(X\), and let \(x\in X\) be a point. Then the following are equivalent:
\begin{enumerate}
\item \(D\equiv_x D'\),
\item \(\Delta_{\widetilde{Y}_{\bullet}}(f^*D)=\Delta_{\widetilde{Y}_{\bullet}}(f^*D')\) for every admissible flag \(\widetilde{Y}_{\bullet}\) centered at \(x\) defined on a smooth projective variety \(\widetilde{X}\) with birational morphism \(f:\widetilde{X}\rightarrow X\),
\item \(\Delta_{\widetilde{Y}_{\bullet}}(f^*D)=\Delta_{\widetilde{Y}_{\bullet}}(f^*D')\) for every proper admissible flag \(\widetilde{Y}_{\bullet}\) over \(X\) centered at \(x\) defined on a smooth projective variety \(\widetilde{X}\) with birational morphism \(f:\widetilde{X}\rightarrow X\),
\item \(\Delta_{\widetilde{Y}_{\bullet}}(f^*D)=\Delta_{\widetilde{Y}_{\bullet}}(f^*D')\) for every infinitesimal flag \(\widetilde{Y}_{\bullet}\) over \(X\) centered at \(x\) defined on a smooth projective variety \(\widetilde{X}\) with birational morphism \(f:\widetilde{X}\rightarrow X\).
\end{enumerate}
{Theorem B.} Let \(D\) and \(D'\) be a big divisors on a smooth projective variety \(X\). For a fixed point \(x\in X\), consider the decompositions \[D=P+N_x^{\mathrm{sm}}+N_X^{\mathrm{sing}}+N_x^c, \quad D'=P'+{N'}_x^{\mathrm{sm}}+{N'}_x^{\mathrm{sing}}+{N'}_x^c.\] Then the following are equivalent
\begin{enumerate}
\item \(P\equiv P'\), \(N_x^{\mathrm{sm}}={N'}_x^{\mathrm{sm}}\), and \(\Delta_{Y_{\bullet}}(N_x^{\mathrm{sing}})=\Delta_{Y_{\bullet}}({N'}_x^{\mathrm{sing}})\) for every admissible flag \(Y_{\bullet}\) centered at \(x\),
\item \(\Delta_{Y_{\bullet}}(D)=\Delta_{Y_{\bullet}}(D')\) for every admissible flag \(Y_{\bullet}\) on \(X\) centered at \(x\),
\item \(\Delta_{\widetilde{Y}_{\bullet}}(f^*D)=\Delta_{\widetilde{Y}_{\bullet}}(f^*D')\) for every induced proper admissible flag \(\widetilde{Y}_{\bullet}\) on \(X\) centered at \(x\) defined on a smooth projective variety \(\widetilde{X}\) with birational morphism \(f:\widetilde{X}\rightarrow X\),
\item \(\Delta_{\widetilde{Y}_{\bullet}}(f^*D)=\Delta_{\widetilde{Y}_{\bullet}}(f^*D')\) for almost every induced infinitesimal admissible flag \(\widetilde{Y}_{\bullet}\) on \(X\) centered at \(x\) defined on a smooth projective variety \(\widetilde{X}\) with birational morphism \(f:\widetilde{X}\rightarrow X\).
\end{enumerate}
Reviewer: Justyna Szpond (Kraków)Families of pointed toric varieties and degenerationshttps://zbmath.org/1496.140342022-11-17T18:59:28.764376Z"Di Rocco, Sandra"https://zbmath.org/authors/?q=ai:di-rocco.sandra"Schaffler, Luca"https://zbmath.org/authors/?q=ai:schaffler.lucaA toric degeneration of an algebraic variety \(X\) is a flat family \(\mathcal{F}\rightarrow \mathbb{A}^1\) whose fiber \(\mathcal{F}_t\) over all points \(t\in\mathbb{A}^1\setminus\{0\}\) are isomorphic to \(X^t\) and whose fiber \(\mathcal{F}_0\) over \(0\) is toric variety \(Y\). Studying degenerations is a standard approach in algebraic geometry. The reason is that many important algebraic invariants of \(X\) coincide with those of \(Y\), e.g. the Hilbert polynomial. In general toric varieties are easier to study than other types of varieties because there is a number of discrete geometric objects, such as polyhedral fans and polytopes associated to them. These objects can be studied with combinatorial methods.
There are two questions which are central to using degenerations. The first one is: how to obtain a toric degeneration of a given variety. And the second is what are the relations between various degenerations of the same variety?
The article under review focuses on the second of questions mentioned above through studying families of toric varieties and their moduli. The specific cases studied here are rather special and the whole article is rather technical. The approach taken is modelled on the construction of the Losev-Manin moduli space parametrizing chains of projective lines with marked points. Genearlizing the idea of twisted Cayley sum, the authors extend Losev-Manin construction to certain families of pointed toric varieties.
Reviewer: Justyna Szpond (Kraków)Correction to: ``Intersection bodies of polytopes''https://zbmath.org/1496.140582022-11-17T18:59:28.764376Z"Berlow, Katalin"https://zbmath.org/authors/?q=ai:berlow.katalin"Brandenburg, Marie-Charlotte"https://zbmath.org/authors/?q=ai:brandenburg.marie-charlotte"Meroni, Chiara"https://zbmath.org/authors/?q=ai:meroni.chiara"Shankar, Isabelle"https://zbmath.org/authors/?q=ai:shankar.isabelleTheorem 2.6 in the authors' paper [ibid. 63, No. 2, 419--439 (2022; Zbl 1493.14101)] is corrected. However, all statements in the original article remain correct.Riemannian submersions of \(\mathrm{SO}_0(2,1)\)https://zbmath.org/1496.220132022-11-17T18:59:28.764376Z"Byun, Taechang"https://zbmath.org/authors/?q=ai:byun.taechang\textit{U. Pinkall} [Invent. Math. 81, 379--386 (1985; Zbl 0585.53051)] proved that, for the Hopf bundle \(S^1\to S^3\to S^2\), the parallel displacement along a simple closed curve in the base space depends only on the area surrounded by the curve. The paper under review studies similar question for the Riemannian submersions \(G \to N\backslash G\), \(G \to A\backslash G\), \(G \to K\backslash G\), and \(G \to NA\backslash G\), where \(G=NAK\) is the Iwasawa decomposition of the Lie group \(G=\mathrm{SO}^0(2,1)\).
Reviewer: Anton Galaev (Hradec Králové)A refinement of Newton and Maclaurin inequalities through abstract convexityhttps://zbmath.org/1496.260182022-11-17T18:59:28.764376Z"Tinaztepe, Gültekin"https://zbmath.org/authors/?q=ai:tinaztepe.gultekin"Tinaztepe, Ramazan"https://zbmath.org/authors/?q=ai:tinaztepe.ramazanSummary: In this study, the refinements of Maclaurin's and Newton's inequalities are given. These refinements are obtained by applying the results on optimality conditions of abstract convex functions. When doing this, we obtain lower bounds for the solutions of some special rational equations.Hypercontractivity and lower deviation estimates in normed spaceshttps://zbmath.org/1496.460062022-11-17T18:59:28.764376Z"Paouris, Grigoris"https://zbmath.org/authors/?q=ai:paouris.grigoris"Tikhomirov, Konstantin"https://zbmath.org/authors/?q=ai:tikhomirov.konstantin-e"Valettas, Petros"https://zbmath.org/authors/?q=ai:valettas.petrosSummary: We consider the problem of estimating small ball probabilities \(\mathbb{P}\{f(G)\le \delta \mathbb{E}f(G)\}\) for subadditive, positively homogeneous functions \(f\) with respect to the Gaussian measure. We establish estimates that depend on global parameters of the underlying function, which take into account analytic and statistical measures, such as the variance and the \({L^1}\)-norms of its partial derivatives. This leads to dimension-dependent bounds for small ball and lower small deviation estimates for seminorms when the linear structure is appropriately chosen to optimize the aforementioned parameters. Our bounds are best possible up to numerical constants. In all regimes, \(\| G\|_{\infty}=\max_{i\le n} |g_i|\) arises as an extremal case in this study. The proofs exploit the convexity and hypercontractivity properties of the Gaussian measure.Reinhardt free spectrahedrahttps://zbmath.org/1496.471282022-11-17T18:59:28.764376Z"McCullough, Scott"https://zbmath.org/authors/?q=ai:mccullough.scott-a"Tuovila, Nicole"https://zbmath.org/authors/?q=ai:tuovila.nicoleSummary: Free spectrahedra are natural objects in the theories of operator systems and spaces and completely positive maps. They also appear in various engineering applications. In this paper, free spectrahedra satisfying a Reinhardt symmetry condition are characterized graph theoretically. It is also shown that, for a simple class of such spectrahedra, automorphisms are trivial.On optimal solutions of well-posed problems and variational inequalitieshttps://zbmath.org/1496.490092022-11-17T18:59:28.764376Z"Ram, Tirth"https://zbmath.org/authors/?q=ai:ram.tirth"Kim, Jong Kyu"https://zbmath.org/authors/?q=ai:kim.jong-kyu|kim.jongkyu"Kour, Ravdeep"https://zbmath.org/authors/?q=ai:kour.ravdeepThe authors consider two problems connected with variational inequalities. In Section 3 they provide a theorem characterising Tykhonov well-posed problems in locally convex topological vector spaces. Further, in section 4, they study variational inequalities in Hilbert spaces. All results are supplied with simple, illustrative examples.
Reviewer: Michał Bełdziński (Łódź)A solution to two old problems by Menger concerning angle spaceshttps://zbmath.org/1496.510052022-11-17T18:59:28.764376Z"Prieto-Martínez, Luis Felipe"https://zbmath.org/authors/?q=ai:prieto-martinez.luis-felipeSummary: Around 1930, Menger expressed his interest in the concept of abstract angle function. He introduced a general definition of this notion for metric and semi-metric spaces. He also proposed two problems concerning conformal embeddability of spaces endowed with an angle function into Euclidean spaces. These problems received attention in later years but only for some particular cases of metric spaces. In this article, we first update the definition of angle function to apply to the larger class of spaces with a notion of betweenness, which seem to us a more natural framework. In this new general setting, we solve the two problems proposed by Menger.Asymptotic geometric analysis. Part IIhttps://zbmath.org/1496.520012022-11-17T18:59:28.764376Z"Artstein-Avidan, Shiri"https://zbmath.org/authors/?q=ai:artstein-avidan.shiri"Giannopoulos, Apostolos"https://zbmath.org/authors/?q=ai:giannopoulos.apostolos-a"Milman, Vitali D."https://zbmath.org/authors/?q=ai:milman.vitali-dThis book is the continuation of the excellent monograph [\textit{S. Artstein-Avidan} et al., Asymptotic geometric analysis. I. Providence, RI: American Mathematical Society (AMS) (2015; Zbl 1337.52001)]. In this series of two books the modern theory of Asymptotic Geometric Analysis is presented. This theory had its origin in the field of (infinite dimensional) Functional Analysis, and evolved largely to a finite dimensional theory, but where the dimension of the objects of study (normed spaces, convex bodies, convex functions...) is very high, increasing to infinity. Both monographs are outstanding and essential references for the study of this theory.
The first four chapters are the natural continuation of the first volume. Thus, in Chapter 1 the authors revisit and deepen the concentration of measure phenomenon (cf. Chapter 3 of Part I), including its relation with important functional inequalities: First, Poincaré's inequality and its role in concentration is studied, and a proof of it in the Gaussian space based on the Ornstein-Uhlenbeck semigroup is provided; also the concentration on the discrete cube is exhaustively analyzed. Then, cost-induced transforms are discussed, including inequalities such as (Weak) Cost-Santaló inequality. I would like to mention another major inequality connected to this question of concentration, the logarithmic Sobolev inequality, which is also studied thoroughly.
Chapter 2 is devoted to discussing major results and problems on isotropic log-concave probability measures, being a follow-up of Chapter 10 in Part I. Among others, the authors deal with the famous Kannan-Lovász-Simonovits conjecture, its equivalent formulation saying that Poincaré's inequality holds for every isotropic log-concave probability measure on \(\mathbb{R}^n\) with a constant independent of the measure or the dimension, or the central limit problem. Important and breakthrough works on these questions by e.g. E. Milman, Eldan, Klartag or Chen are described.
The Gaussian distribution is one of the cornerstones in probability theory, and Chapter 3 deals with some fundamental isoperimetric inequalities about the \(n\)-dimensional Gaussian measure \(\gamma_n\) (cf. Chapter 9 of Part I). Naturally, one of the main results in this chapter is the engaging form of the isoperimetric inequality stating that if \(A\) is a Borel set in \(\mathbb{R}^n\) and \(H\) is a half-space with \(\gamma_n(A)=\gamma_n(H)\), then \(\gamma_n(A_t)\geq\gamma_n(H_t)\) for all \(t>0\), being \(A_t\) the \(t\)-extension of \(A\). Three different proofs of this important fact are presented (based on reducing it to the isoperimetric problem for the sphere, on a functional inequality or on a Gaussian symmetrization). Other significant results on this topic are studied here, such as Ehrhard's inequality, the behavior of the Gaussian measure with respect to dilates of a centrally symmetric convex body, the Gaussian correlation inequality, or the \(B\)-theorem of Cordero-Erausquin, Fradelizi and Maurey. Some applications of geometric inequalities for the Gaussian measure to discrepancy problems are also presented.
In Chapter 4, different volume-type inequalities are studied, continuing so the analysis that was made in Chapters 2 and 10 of Part I. First, the Brascamp-Lieb-Luttinger inequality, and the multidimensional versions of the Brascamp-Lieb and the Barthe inequalities are discussed, and several applications of these thorough results to different problems in Convex Geometry are presented. We highlight the Gluskin-Milman theorem, which is applied to show that every \(n\)-dimensional normed space has the random cotype-2 property. Next, volume estimates for convex bodies with few vertices or facets are exposed, as Vaaler's inequality bounding from below the volume of the intersection of a finite number of origin-symmetric strips, and other related results. Shephard's problem is also discussed, as well as its (strongly) negative answers by Petty and Schneider, and by Ball. Then the authors outline a theory that has been developed in several works of Paouris, Pivovarov et al., which provides a unified way of showing well-known inequalities from geometric probability. The chapter concludes considering the Blaschke-Petkantschin formulae and their geometric applications. Among them, Giannopoulos-Koldobsky's positive answer to a variant of the Busemann-Petty (and Shephard) problem, proposed by Milman, is presented.
Chapter 5 is devoted to the delightful theory of type and cotype, introduced and developed mainly by Maurey and Pisier. It starts with several basic notions and facts, and continues discussing the absolutely summing operators, nuclear operators, trace duality, the Gaussian type and cotype and the \(\ell\)-norm. Then, some developments on the duality of entropy problem for spaces with type \(p\) are analyzed, as well as some results for spaces with bounded cotype constant: the so-called ``Maurey-Pisier lemma'' and a theorem by Bourgain and Milman asserting that the volume ratio of the unit ball of an \(n\)-dimensional normed space is bounded by a function of the cotype 2 constant of the space. The last half of the chapter focuses on: Grothendieck's inequality; Kwapien's theorem asserting that the (Banach-Mazur) distance from a Banach space \(X\) to some Hilbert space is bounded from above by the product of the type 2 and the cotype 2 constants of \(X\); a theorem of Lindenstrauss and Tzafriri providing a positive answer to the complemented subspace problem; and Krivine's theorem and a counterpart/strengthening of it by Maurey and Pisier. A result of Johnson and Schechtman about embedding \(\ell^m_p\) into \(\ell^n_1\) closes the chapter.
Next, in Chapter 6 the geometry of the family of all normed (\(n\)-dimensional) spaces equipped with the Banach-Mazur distance (i.e., the so-called Banach-Mazur compactum) is investigated. The question of computing the diameter of the compactum is the starting point of the chapter, and Gluskin's theorem is shown: there exists an absolute constant \(c>0\) such that, for any \(n\in\mathbb{N}\), there exist two \(n\)-dimensional normed spaces with distance greater than \(cn\). Also the problem to estimate the diameter of the compactum in the non-symmetric case is discussed (the best known upper bound is due to Rudelson), for which the method of random orthogonal factorizations is previously introduced, as well as several applications of it. Next the authors consider Pełcynski's question about the asymptotic growth of the radius of the Banach-Mazur compactum with respect to \(\ell^n_{\infty}\); this problem is still open, the best known upper bound due to Giannopoulos being \(O(n^{5/6})\). The chapter ends with the study of several results from the local theory of normed spaces: Elton's theorem; a result by Milman-Wolfson about spaces with maximal distance to the Euclidean space; Alon-Milman's theorem on the existence of \(k\)-dimensional subespaces of a normed space with small distance to either \(\ell^k_2\) or \(\ell^k_{\infty}\); and the Schechtman theorem on the dependence on \(\varepsilon\) of the critical dimension in Dvoretzky theorem.
Symmetrizations of sets are one of the cornerstones in Convexity, and a major question in this matter is knowing how fast a sequence of symmetrizations approaches an Euclidean ball starting from an arbitrary convex body. Chapter 7 focuses on this issue, departing from two important symmetrizations: the Steiner and the Minkowski symmetrizations. While the first one has been most commonly studied in other monographs (although not from the point of view here considered), it is not the case for the second one, so conferring an extra incentive to the reading of this chapter. It is worthwhile to stress that in order to achieve this goal, the authors use tools from Asymptotic Geometric Analysis rather than the methods in classical Convex Geometry, which allow to show that the number of required symmetrizations in order to get ``close'' to an Euclidean ball is linear in the dimension. Works of Bourgain, Lindenstrauss and Milman, and mainly of Klartag, on this regard are presented, for both the Minkowski and the Steiner symmetrizations.
Next, Chapter 8 deals with the method of interlacing families of polynomials, focusing on its applications to Geometric Functional Analysis and Convex Geometry. It starts with a result of Batson, Spielman and Srivastava asserting, geometrically speaking, that a John decomposition of the identity can be approximated by a John sub-decomposition with suitable weights, involving a linear (in the dimension) number of terms. Then the interlacing polynomials are studied, and used afterwards for proving the restricted invertibility principle; other forms and generalizations of this theorem are also considered, one of which allows to show the proportional Dvoretzky-Rogers factorization theorem. The chapter ends with a full overview of the Kadison-Singer problem.
The book concludes with a wonderful Chapter 9, entitled ``Functionalization of Geometry'', where several sorts of functions on \(\mathbb{R}^n\) are studied from a geometric point of view: among them, of course, log-concave functions are the leading ones. Their importance in this context lies in the fact that they can be seen as an extension of (closed) convex sets, and particular operations between them as fundamental constructions in convex geometry; as an outcome, every geometric inequality has an analytic counterpart. Thus, after a first section devoted to log-concavity, functional duality is studied and, among others, it is proved that the Legendre transform is, up to linear variants, the only transform \(T\) defined on the set of convex functions on \(\mathbb{R}^n\) satisfying that \(T\circ T=Id\) and that \(\varphi\leq\psi\) if and only if \(T\varphi\geq T\psi\). The chapter ends showing functional versions of several fundamental geometric inequalities: Brunn-Minkowski, Uryshon, Blaschke-Santaló, reverse Brunn-Minkowski and Blaschke-Santaló, Rogers-Shephard, etc.
In line with the first volume of this series, all chapters are enriched with a final section, a collection of ``Notes and remarks'', where the main references regarding the content of the chapter can be found, as well as applications and many other problems related to its subject. The bibliography is large and exhaustive, consisting of almost 750 items, and including both, classical and very recent and seminal references.
This series of two books will most certainly be a fundamental source for the study and investigation in the field of Asymptotic Geometric Analysis.
Reviewer: Maria A. Hernández Cifre (Murcia)A unified approach to collectively maximal elements in abstract convex spaceshttps://zbmath.org/1496.520022022-11-17T18:59:28.764376Z"Park, Sehie"https://zbmath.org/authors/?q=ai:park.sehieThe author establishes a very KKM type theorem in abstract convex spaces from which he then obtains an abstract collectively maximal element theorem. Finally, the author shows that a large number of previous theorems on the existence of maximal element and of equilibrium can be derived from his result.
Reviewer: Mircea Balaj (Oradea)Smallest equiprojective polyhedronhttps://zbmath.org/1496.520032022-11-17T18:59:28.764376Z"Hasan, Masud"https://zbmath.org/authors/?q=ai:hasan.masud"Hossain, Mohammad Monoar"https://zbmath.org/authors/?q=ai:hossain.mohammad-monoar"López-Ortiz, Alejandro"https://zbmath.org/authors/?q=ai:lopez-ortiz.alejandro"Nusrat, Sabrina"https://zbmath.org/authors/?q=ai:nusrat.sabrinaSummary: A convex polyhedron \(P\) is \(k\)-equiprojective if all of its orthogonal projections, i.e., shadows, except those parallel to the faces of \(P\) are \(k\)-gons for some fixed value of \(k\). Since 1968 it is an open problem to construct all equiprojective polyhedra. In this paper, we prove that there is no 3- or 4-equiprojective polyhedron and a triangular prism is the only 5-equiprojective polyhedron (thus the smallest equiprojective polyhedron).Topological and geometric properties of the set of 1-nonconvexity points of a weakly 1-convex set in the planehttps://zbmath.org/1496.520042022-11-17T18:59:28.764376Z"Osipchuk, T. M."https://zbmath.org/authors/?q=ai:osipchuk.tatyana-mikhailovna|osipchuk.tetiana-mSummary: We consider a class of generalized convex sets in the real plane known as weakly 1-convex sets. For a set in the real Euclidean space \(\mathbb{R}^n\), \(n \geq 2,\) we say that a point of the complement of this set to the entire space \(\mathbb{R}^n\) is an \textit{m-nonconvexity point of the set}, \( m=\overline{1,n-1} \), if any \(m\)-dimensional plane passing through this point crosses the indicated set. An open set in the space \(\mathbb{R}^n\), \(n \geq 2,\) is called \textit{weakly m-convex}, \( m=\overline{1,n-1} \), if its boundary does not contain any \(m\)-nonconvexity points of the set. Moreover, in the class of open weakly 1-convex sets in the plane, we select a subclass of sets with finitely many connected components and a nonempty set of 1-nonconvexity points. We mainly analyze the properties of the set of 1-nonconvexity points for the sets from the indicated subclass. In particular, for any set in this subclass, it is proved that the set of its 1-nonconvexity points is open, that any connected component of the set of its 1-nonconvexity points is the interior of a convex polygon, and that, for any convex polygon, there exists a set from the indicated subclass such that its set of 1-nonconvexity points coincides with the interior of a polygon.On a version of the slicing problem for the surface area of convex bodieshttps://zbmath.org/1496.520052022-11-17T18:59:28.764376Z"Brazitikos, Silouanos"https://zbmath.org/authors/?q=ai:brazitikos.silouanos"Liakopoulos, Dimitris-Marios"https://zbmath.org/authors/?q=ai:liakopoulos.dimitris-mariosSummary: We study the slicing inequality for the surface area instead of volume. This is the question whether there exists a constant \(\alpha_n\) depending (or not) on the dimension \(n\) so that
\[S(K)\leqslant \alpha_n|K|^{\frac{1}{n}}\max_{\xi \in S^{n-1}}S(K\cap \xi^{\perp }),\]
where \(S\) denotes surface area and \(|\cdot |\) denotes volume. For any fixed dimension we provide a negative answer to this question, as well as to a weaker version in which sections are replaced by projections onto hyperplanes. We also study the same problem for sections and projections of lower dimension and for all the quermassintegrals of a convex body. Starting from these questions, we also introduce a number of natural parameters relating volume and surface area, and provide optimal upper and lower bounds for them. Finally, we show that, in contrast to the previous negative results, a variant of the problem which arises naturally from the surface area version of the equivalence of the isomorphic Busemann-Petty problem with the slicing problem has an affirmative answer.The metric projections onto closed convex cones in a Hilbert spacehttps://zbmath.org/1496.520062022-11-17T18:59:28.764376Z"Qiu, Yanqi"https://zbmath.org/authors/?q=ai:qiu.yanqi"Wang, Zipeng"https://zbmath.org/authors/?q=ai:wang.zipengSummary: We study the metric projection onto the closed convex cone in a real Hilbert space \(\mathscr{H}\) generated by a sequence \(\mathcal{V} = \{v_n\}_{n=0}^\infty\). The first main result of this article provides a sufficient condition under which the closed convex cone generated by \(\mathcal{V}\) coincides with the following set:
\[
\mathcal{C}[[\mathcal{V}]]: = \bigg\{\sum_{n=0}^\infty a_n v_n\Big|a_n\geq 0, \text{ the series }\sum_{n=0}^\infty a_n v_n \text{ converges in } \mathscr{H}\bigg\}.
\]
Then, by adapting classical results on general convex cones, we give a useful description of the metric projection onto \(\mathcal{C}[[\mathcal{V}]]\). As an application, we obtain the best approximations of many concrete functions in \(L^2([-1,1])\) by polynomials with nonnegative coefficients.Strongly-Delaunay starshaped polygonshttps://zbmath.org/1496.520072022-11-17T18:59:28.764376Z"Bloch, Ethan D."https://zbmath.org/authors/?q=ai:bloch.ethan-dSummary: In the course of his study of simplexwise linear maps of disks, Ho gave conditions under which a single vertex of a starshaped polygon in the plane can be moved in such a way that the polygon stays starshaped. We show that the analog of this result does not hold with the added condition that for each starshaped polygon throughout the move there is a cone triangulation of the starshaped polygon that is a strongly-Delaunay triangulation. Nonetheless, we show that the space of all orientation-preserving starshaped polygons that have strongly-Delaunay cone triangulations has the homotopy type of \(S^1\).Volume properties and some characterizations of ellipsoids in \(\mathbb{E}^{n+1}\)https://zbmath.org/1496.520082022-11-17T18:59:28.764376Z"Kim, Dong-Soo"https://zbmath.org/authors/?q=ai:kim.dongsoo-s"Kim, Incheon"https://zbmath.org/authors/?q=ai:kim.incheon"Kim, Young Ho"https://zbmath.org/authors/?q=ai:kim.younghoSummary: Suppose that \(M\) is a strictly convex and closed hypersurface in \(\mathbb{E}^{n+1}\) with the origin \(o\) in its interior. We consider the homogeneous function \(g\) of positive degree \(d\) satisfying \(M=g^{-1}(1)\). Then, for a positive number \(h\) the level hypersurface \(g^{-1}(h)\) of \(g\) is a homothetic hypersurface of \(M\) with respect to the origin \(o\). In this paper, for tangent hyperplanes \(\Phi_h\) to \(g^{-1}(h)\) (\(0 < h < 1\)), we study the \((n + 1)\)-dimensional volume of the region enclosed by \(\Phi_h\) and the hypersurface \(M\), etc. As a result, with the aid of the theorem of Blaschke and Deicke for proper affine hypersphere centered at the origin, we establish some characterizations for ellipsoids in \(\mathbb{E}^{n+1}\). As a corollary, we extend Schneider's characterization for ellipsoids in \(\mathbb{E}^3\). Finally, for further study, we raise a question for elliptic paraboloids which was originally conjectured by Golomb.Unique determination of ellipsoids by their dual volumeshttps://zbmath.org/1496.520092022-11-17T18:59:28.764376Z"Myroshnychenko, Sergii"https://zbmath.org/authors/?q=ai:myroshnychenko.sergii"Tatarko, Kateryna"https://zbmath.org/authors/?q=ai:tatarko.kateryna"Yaskin, Vladyslav"https://zbmath.org/authors/?q=ai:yaskin.vladyslavSummary: Gusakova and Zaporozhets conjectured that ellipsoids in \(\mathbb{R}^n\) are uniquely determined (up to an isometry) by their intrinsic volumes. Petrov and Tarasov confirmed this conjecture in \(\mathbb{R}^3\). In this paper, we solve the dual problem in all dimensions. We show that any ellipsoid in \(\mathbb{R}^n\) centered at the origin is uniquely determined (up to an isometry) by an \(n\)-tuple of its dual volumes. As an application, we give an alternative proof of the result of Petrov and Tarasov.The equilateral small octagon of maximal widthhttps://zbmath.org/1496.520102022-11-17T18:59:28.764376Z"Bingane, Christian"https://zbmath.org/authors/?q=ai:bingane.christian"Audet, Charles"https://zbmath.org/authors/?q=ai:audet.charlesLet \(P\) be the set of convex polygons in \(\mathbb{R}^2\) with diameter equals to 1. The maximal width of an equilateral \(C\in P\) with \(2^s\) sides is unknown for \(s \geq 3\). The authors solve the case \(s=3\) and present an optimal 8-gon. They also propose a lower bound on the maximal width for \(s\geq 4\) by constructing a family of equilateral \(2^s\)-gons whose widths are between the width of regular \(2^s\)-gon and such a bound. The paper includes numerical computations and a lot of figures of polygons that help the reader.
Reviewer: Pedro Martín Jiménez (Badajoz)On mixed Hodge-Riemann relations for translation-invariant valuations and Aleksandrov-Fenchel inequalitieshttps://zbmath.org/1496.520112022-11-17T18:59:28.764376Z"Kotrbatý, Jan"https://zbmath.org/authors/?q=ai:kotrbaty.jan"Wannerer, Thomas"https://zbmath.org/authors/?q=ai:wannerer.thomasProperties of a curve whose convex hull covers a given convex bodyhttps://zbmath.org/1496.520122022-11-17T18:59:28.764376Z"Nikonorov, Yurii G."https://zbmath.org/authors/?q=ai:nikonorov.yurii-gSummary: In this note, we prove the following inequality for the norm \(N(K)\) of a convex body \(K\) in \(\mathbb{R}^n, n\geq 2\):
\[
N(K) \leq \frac{\pi^{\frac{n-1}{2}}}{2 \Gamma \left( \frac{n+1}{2}\right)}\cdot\,\text{length}\,(\gamma )+\frac{\pi^{\frac{n}{2}-1}}{\Gamma \left( \frac{n}{2}\right)} \cdot\,\text{diam}\,(K),
\]
where \(\text{diam}(K)\) is the diameter of \(K, \gamma\) is any curve in \(\mathbb{R}^n\) whose convex hull covers \(K\), and \(\Gamma\) is the gamma function. If in addition \(K\) has constant width \(\Theta\), then we get the inequality
\[
\text{length}\,(\gamma ) \geq \frac{2(\pi -1)\Gamma \left( \frac{n+1}{2}\right)}{\sqrt{\pi}\,\Gamma \left( \frac{n}{2}\right)}\cdot \Theta \geq 2(\pi -1) \cdot \sqrt{\frac{n-1}{2\pi}}\cdot \Theta.
\]
In addition, we pose several unsolved problems.Symmetrized Talagrand inequalities on Euclidean spaceshttps://zbmath.org/1496.520132022-11-17T18:59:28.764376Z"Tsuji, Hiroshi"https://zbmath.org/authors/?q=ai:tsuji.hiroshi.1From the Introduction: The Talagrand inequality, which is also called the Talagrand transportation inequality, is as follows: If \(m = e^{-V}\, L_n\) is a probability measure on \(\mathbb{R}^n\) with \(\nabla^2V\ge k\) for some \(k>0\), and \(\mu\in P_2(\mathbb{R}^n)\), then \(W_2^2(\mu, m) \le \mathrm{Ent}_m(\mu)/k\) holds, where \(L_n\) is the Lebesgue measure on \(\mathbb{R}^n\), \(P_2(\mathbb{R}^n)\) is the set of all probability measures on \(\mathbb{R}^n\) with finite second moment, \(W_2\) is the Wasserstein distance, and \(\mathrm{Ent}_m\) is the relative entropy (or the Kullback-Leibler distance) with respect to \(m\). More generally, it is known that the Talagrand inequality holds on metric measure spaces with similar conditions to those above, and there are many studies on refinements of the Talagrand inequality and relations with logarithmic Sobolev inequalities and Poincaré inequalities [\textit{C. Villani}, Optimal transport. Old and new. Berlin: Springer (2009; Zbl 1156.53003)]. This paper is motivated by \textit{M. Fathi}'s following result [Electron. Commun. Probab. 23, Paper No. 81, 9 p. (2018; Zbl 1400.35008)]:
Theorem 1. Let \(\mu\), \(\nu\in P_2(\mathbb{R}^n)\). The following assertions hold:
\begin{itemize}
\item[(\textbf{1})] Let \(m = e^{-V}\, L_n\) be a probability measure on \(\mathbb{R}^n\) such that \(V\in C^{\infty}(\mathbb{R}^n)\) is even and \(k\)-convex for some \(k>0\). If \(\nu\) is symmetric (i.e., its density with respect to \(L_n\) is even), then it holds that \(1/2\, W_2^2(\mu, \nu) \le 1/k (\mathrm{Ent}_m(\mu) + \mathrm{Ent}_m(\nu))\).
\item[(\textbf{2})] Let \(m=\gamma_n\) be the \(n\)-dimensional standard Gaussian measure. If \(\int_{\mathbb{R}^n}x\, d\nu(x)=0\), then it holds that \(1/2\, W_2^2(\mu, \nu) \le \mathrm{Ent}_{\gamma_n}(\mu) + \mathrm{Ent}_{\gamma_n}(\nu))\).
\end{itemize}
Moreover, the equality holds in (\textbf{2}) if and only if there exist some positive definite symmetric matrix \(A\in \mathbb{R}^{n\times n}\) and some \(a\in \mathbb{R}\) such that \(\mu\) is the Gaussian measure whose center is \(a\) and covariance matrix is \(A\), and \(\nu\) is the Gaussian measure whose center is \(0\) and covariance matrix is \(A^{-1}\).
Note that Theorem 1 (\textbf{1}) does not include (\textbf{2}). When \(m=\nu\) in (\textbf{1}), we recover the classical Talagrand inequality, and hence Theorem 1 is a refinement of the classical Talagrand inequality provided \(V\) is even. Using Fathi's paper as reference, the author of the present paper calls this type of inequality a symmetrized Talagrand inequality. M. Fathi proved the symmetrized Talagrand inequality by using optimal transport theory and convex geometry. Moreover, M. Fathi pointed out that the symmetrized Talagrand inequality for Gaussian measures is related to the functional Blaschke-Santalo inequality, which is well known and important in convex geometry. This paper considers refinements and extensions of the symmetrized Talagrand inequality.
The present paper is organized as follows. Some fundamental notions from optimal transport theory and functional inequalities are introduced, including Talagrand inequalities. Section 3 gives another form of the symmetrized Talagrand inequality by a self-improvement of Fathi's result (Theorem 1 (\textbf{2})). The barycenter of a probability measure plays an important role in this section. In Section 4, the author proves the main theorems and apply them to prove the corresponding \(HWI\) inequalities (between entropy \(H\), Wasserstein distance \(W\) and Fisher information \(I\)), logarithmic Sobolev inequalities and Poincaré inequalities. Moreover, the author gives an alternative proof of Theorem 1 and an extension on the real line in Section 4.3. In the final section, the author describes an application of the result in the previous subsection to the Blaschke-Santalo inequality for log-concave probability measures.
Reviewer: Viktor Ohanyan (Erevan)Some non-trivial examples of equiprojective polyhedrahttps://zbmath.org/1496.520142022-11-17T18:59:28.764376Z"Hasan, Masud"https://zbmath.org/authors/?q=ai:hasan.masud"Hossain, Mohammad Monoar"https://zbmath.org/authors/?q=ai:hossain.mohammad-monoar"López-Ortiz, Alejandro"https://zbmath.org/authors/?q=ai:lopez-ortiz.alejandro"Nusrat, Sabrina"https://zbmath.org/authors/?q=ai:nusrat.sabrina"Quader, Saad Altaful"https://zbmath.org/authors/?q=ai:quader.saad-altaful"Rahman, Nabila"https://zbmath.org/authors/?q=ai:rahman.nabilaA \(3\)-dimensional Euclidean polytope \(P\) is equiprojective when its \(2\)-dimensional projections along all possible directions (except for the directions parallel to a facet of \(P\)) are polygons with the same number of edges. If these polygons all have \(k\) edges, then \(P\) is called \(k\)-equiprojective. A question of \textit{G. C. Shephard} [Math. Gaz. 52, 359--367 (1968; Zbl 0167.20303)] asks for a description of all possible equiprojective polytopes. The question is still open, though a practical criterion that allows to recognise equiprojective polytopes has been given by \textit{M. Hasan} and \textit{A. Lubiw} [Comput. Geom. 40, No. 2, 148--155 (2008; Zbl 1137.52001)].
In this article, five combinatorial types of equiprojective polytopes are provided by truncating Johnson solids and an infinite family of combinatorial types by glueing two prisms along a facet. The Johnson solids are the \(3\)-dimensional polytopes whose facets all are regular polygons. There are \(92\) such polytopes. The Johnson solids that the authors truncate in order to construct equiprojective polytopes are
\begin{itemize}
\item[(i)] the regular tetrahedron (that is transformed into a \(10\)-equiprojective polytope after \(7\) truncations),
\item[(ii)] the pyramid over a square (that is also transformed into a \(10\)-equiprojective polytope using a sequence of \(5\) truncations),
\item[(iii)] the triangular cupola (that is turned into a \(11\)-equiprojective polytope using a sequence of \(5\) truncations),
\item[(iv)] the pentagonal rotunda (that is transformed into a \(21\)-equiprojective polytope and a \(23\)-equiprojective polytope using two different sequences of truncations).
\end{itemize}
The construction of an infinite family of equiprojective polytopes goes as follows. Consider a regular polygon with \(2k-2\) vertices and cut it along the line segment between two opposite vertices. This results in two identical polygons with \(k\) vertices. Denote by \(p(k)\) that polygon and by \(e\) its edge along which the cut was performed. Now consider a prism \(P(k)\) over \(p(k)\). In that prism, \(e\) is incident to \(p(k)\) and to another facet \(f\) that results from taking the prism. The prism can be chosen in such a way that \(f\) is a square. Consider a second prism \(P(k')\) constructed using the same procedure, but possibly with \(k'\neq{k}\). Since \(f\) is a square, there are two ways to glue \(P(k)\) to \(P(k')\) along \(f\): one of them results in a prism over a polygon with \(k+k'-2\) vertices. The other results in a \((k+k')\)-equiprojective polytope.
The authors finally observe that for all \(k\geq6\), this construction provides \(1+\lfloor(k-6)/2\rfloor\) different combinatorial types of \(k\)-equiprojective polytopes.
Reviewer: Lionel Pournin (Paris)On lattice width of lattice-free polyhedra and height of Hilbert baseshttps://zbmath.org/1496.520152022-11-17T18:59:28.764376Z"Henk, Martin"https://zbmath.org/authors/?q=ai:henk.martin"Kuhlmann, Stefan"https://zbmath.org/authors/?q=ai:kuhlmann.stefan"Weismantel, Robert"https://zbmath.org/authors/?q=ai:weismantel.robertEhrhart polynomials of rank two matroidshttps://zbmath.org/1496.520162022-11-17T18:59:28.764376Z"Ferroni, Luis"https://zbmath.org/authors/?q=ai:ferroni.luis"Jochemko, Katharina"https://zbmath.org/authors/?q=ai:jochemko.katharina"Schröter, Benjamin"https://zbmath.org/authors/?q=ai:schroter.benjaminSummary: Over a decade ago De Loera, Haws and Köppe conjectured that Ehrhart polynomials of matroid polytopes have only positive coefficients and that the coefficients of the corresponding \(h^\ast \)-polynomials form a unimodal sequence. The first of these intensively studied conjectures has recently been disproved by the first author who gave counterexamples in all ranks greater than or equal to three. In this article we complete the picture by showing that Ehrhart polynomials of matroids of lower rank have indeed only positive coefficients. Moreover, we show that they are coefficient-wise bounded by the Ehrhart polynomials of minimal and uniform matroids. We furthermore address the second conjecture by proving that \(h^\ast \)-polynomials of matroid polytopes of sparse paving matroids of rank two are real-rooted and therefore have log-concave and unimodal coefficients. In particular, this shows that the \(h^\ast \)-polynomial of the second hypersimplex is real-rooted, thereby strengthening a result of De Loera, Haws and Köppe.\(\mathrm{SL}(n)\) contravariant vector valuationshttps://zbmath.org/1496.520172022-11-17T18:59:28.764376Z"Li, Jin"https://zbmath.org/authors/?q=ai:li.jin.4|li.jin.5|li.jin|li.jin.3|li.jin.2|li.jin.1"Ma, Dan"https://zbmath.org/authors/?q=ai:ma.dan.1|ma.dan"Wang, Wei"https://zbmath.org/authors/?q=ai:wang.wei.16Let \(\phi\in SL(n)\); this may be considered as acting not only on \(\mathbb{R}^n\) but also on the space \(\mathcal{P}^n\) of polytopes in \(\mathbb{R}^n\). This allows a vector-valued valuation \(Z:\mathcal{P}^n\rightarrow\mathbb{R}^n\) to be described as \(SL(n)\)-\textit{covariant} if, for all \(\phi\in SL(n)\), \(Z\phi=\phi Z\). In [Trans. Am. Math. Soc. 370, No. 12, 8999--9023 (2018; Zbl 1406.52031)], \textit{C. Zeng} and \textit{D. Ma} gave a complete characterization of such valuations.
Similarly, \(Z\) is \(SL(n)\)-\textit{contravariant} if, for all \(\phi\in SL(n)\), \(Z\phi=\phi^{-1}Z\). This paper gives a characterization of \(SL(n)\)-contravariant valuations, first for polytopes that contain the origin and then for general polytopes. Interestingly, the two-dimensional cases are significantly more complicated than those in higher dimensions.
Reviewer: Robert Dawson (Halifax)Separating circles on the sphere by polygonal tilingshttps://zbmath.org/1496.520182022-11-17T18:59:28.764376Z"Bezdek, András"https://zbmath.org/authors/?q=ai:bezdek.andrasSummary: We say that a tiling separates discs of a packing in the Euclidean plane, if each tile contains exactly one member of the packing. It is a known elementary geometric problem to show that for each locally finite packing of circular discs, there exists a separating tiling with convex polygons. In this paper we show that this separating property remains true for circle packings on the sphere and in the hyperbolic plane. Moreover, we show that in the Euclidean plane circles are the only convex discs, whose packings with similar copies can be always separated by polygonal tilings. The analogous statement is not true on the sphere and it is not known in the hyperbolic plane.Coverings of planar and three-dimensional sets with subsets of smaller diameterhttps://zbmath.org/1496.520192022-11-17T18:59:28.764376Z"Tolmachev, A. D."https://zbmath.org/authors/?q=ai:tolmachev.a-d"Protasov, D. S."https://zbmath.org/authors/?q=ai:protasov.d-s"Voronov, V. A."https://zbmath.org/authors/?q=ai:voronov.vsevolod-aleksandrovichSummary: Quantitative estimates related to the classical Borsuk problem of splitting set in Euclidean space into subsets of smaller diameter are considered. For a given \(k\) there is a minimal diameter of subsets at which there exists a covering with \(k\) subsets of any planar set of unit diameter. In order to find an upper estimate of the minimal diameter we propose an algorithm for finding sub-optimal partitions. In the cases \(10 \leqslant k \leqslant 17\) some upper and lower estimates of the minimal diameter are improved. Another result is that any set \(M \subset \mathbb{R}^3\) of a unit diameter can be partitioned into four subsets of a diameter not greater than 0.966.Packings with geodesic and translation balls and their visualizations in \(\widetilde{\mathbf{SL}_2\mathbf{R}}\) spacehttps://zbmath.org/1496.520202022-11-17T18:59:28.764376Z"Molnár, Emil"https://zbmath.org/authors/?q=ai:molnar.emil"Szirmai, Jenő"https://zbmath.org/authors/?q=ai:szirmai.jenoSummary: Remembering on our friendly cooperation between the Geometry Departments of Technical Universities of Budapest and Vienna (also under different names) a nice topic comes into consideration: the ``Gum fibre model'' (see Fig. 1).
One point of view is the so-called kinematic geometry by Vienna colleagues, e.g., as in [\textit{H. Stachel}, Math. Appl., Springer 581, 209--225 (2006; Zbl 1100.52005)], but also in very general context. The other point is the so-called \(\mathbf{H}^2 \times \mathbf{R}\) geometry and \(\widetilde{\mathbf{SL}_2\mathbf{R}}\) geometry where -- roughly -- two hyperbolic planes as circle discs are connected with gum fibres, first: in a simple way, second: in a twisted way. This second homogeneous (Thurston) geometry will be our topic (initiated by Budapest colleagues, and discussed also in international cooperations).
We use for the computation and visualization of \(\widetilde{\mathbf{SL}_2\mathbf{R}}\) its projective model, as in our previous papers. We found seemingly extremal geodesic ball packing for \(\widetilde{\mathbf{SL}_2\mathbf{R}}\) group \(\mathbf{pq}_k \mathbf{o}_\ell\) (\(p = 9\), \(q = 3\), \(k = 1\), \(o = 2\), \(\ell = 1\)) with density
\(\approx 0.787758\) (Table 2). Much better translation ball packing is for group \(\mathbf{pq}_k \mathbf{o}_\ell\) (\(p = 11\), \(q = 3\), \(k = 1\), \(o = 2\), \(\ell = 1\)) with density \(\approx 0.845306\) (Table 3).On tilings of asymmetric limited-magnitude ballshttps://zbmath.org/1496.520212022-11-17T18:59:28.764376Z"Wei, Hengjia"https://zbmath.org/authors/?q=ai:wei.hengjia"Schwartz, Moshe"https://zbmath.org/authors/?q=ai:schwartz.mosheSummary: We study whether an asymmetric limited-magnitude ball may tile \(\mathbb{Z}^n\). This ball generalizes previously studied shapes: crosses, semi-crosses, and quasi-crosses. Such tilings act as perfect error-correcting codes in a channel which changes a transmitted integer vector in a bounded number of entries by limited-magnitude errors. A construction of lattice tilings based on perfect codes in the Hamming metric is given. Several non-existence results are proved, both for general tilings, and lattice tilings. A complete classification of lattice tilings for two certain cases is proved.Local structure of karyon tilingshttps://zbmath.org/1496.520222022-11-17T18:59:28.764376Z"Zhuravlev, V. G."https://zbmath.org/authors/?q=ai:zhuravlev.vladimir-gThe paper is devoted to karyon tilings of the torus \(\mathbb{T}^d\) of an arbitrary dimension \(d\). The prototypes of karyon tilings are the one-dimensional Fibonacci tilings and the two-dimensional fractal Rauzy tilings, described earlier by the author.
The set of \(d+1\) vectors in \(\mathbb{R}^d\) is a star if any \(d-1\) its vectors are linearly independent and if the hyperplane determined by these \(d-1\) vectors separates the two remaining vectors. The star is one of the basic concepts in the construction of karyon tilings given in previous works of the author.
In the paper under review, local properties of karyon tilings are studied. In particular, tilings of the torus are classified depending on polyhedral stars and ray stars, the connection between ray and polyhedral stars is established.
Karyon tilings have applcations to multidimensional continued fractions.
Reviewer: Elizaveta Zamorzaeva (Chişinău)Symmetry properties of karyon tilingshttps://zbmath.org/1496.520232022-11-17T18:59:28.764376Z"Zhuravlev, V. G."https://zbmath.org/authors/?q=ai:zhuravlev.vladimir-gKaryon tilings are multidimensional generalizations of the one-dimensional Fibonacci tilings and the two-dimensional Rauzy tilings. In the previous article [J. Math. Sci., New York 264, No. 2, 122--149 (2022; Zbl 1496.52022); translation from Zap. Nauchn. Semin. POMI 502, 32--73 (2021)], the author has investigated local properties of karyon tilings of the torus \(\mathbb{T}^d\) of an arbitrary dimension \(d\).
In the present paper, symmetry properties of karyon tilings of the torus \(\mathbb{T}^d\) are studied. The main results obtained are as follows:
\begin{itemize}
\item[1.] A karyon tiling is invariant with respect to canonical shift of the torus \(\mathbb{T}^d\). This is a fundamental property of karyon tilings. The action of shift consists in exchanging the karyon of the tiling, which is composed of \(d+1\) parallelepipeds.
\item[2.] A nondegenerate karyon tiling has \(2^d\) central symmetries.
\end{itemize}
In general the presence of symmetries is connected with optimality properties. The karyon tilings have applications to multidimensional continued fractions.
Reviewer: Elizaveta Zamorzaeva (Chişinău)Sharing pizza in \(n\) dimensionshttps://zbmath.org/1496.520242022-11-17T18:59:28.764376Z"Ehrenborg, Richard"https://zbmath.org/authors/?q=ai:ehrenborg.richard"Morel, Sophie"https://zbmath.org/authors/?q=ai:morel.sophie"Readdy, Margaret"https://zbmath.org/authors/?q=ai:readdy.margaret-aSummary: We introduce and prove the \(n\)-dimensional Pizza Theorem: Let \(\mathcal{H}\) be a hyperplane arrangement in \(\mathbb{R}^n \). If \(K\) is a measurable set of finite volume, the pizza quantity of \(K\) is the alternating sum of the volumes of the regions obtained by intersecting \(K\) with the arrangement \(\mathcal{H} \). We prove that if \(\mathcal{H}\) is a Coxeter arrangement different from \(A_1^n\) such that the group of isometries \(W\) generated by the reflections in the hyperplanes of \(\mathcal{H}\) contains the map \(-\text{id} \), and if \(K\) is a translate of a convex body that is stable under \(W\) and contains the origin, then the pizza quantity of \(K\) is equal to zero. Our main tool is an induction formula for the pizza quantity involving a subarrangement of the restricted arrangement on hyperplanes of \(\mathcal{H}\) that we call the even restricted arrangement. More generally, we prove that for a class of arrangements that we call even (this includes the Coxeter arrangements above) and for a sufficiently symmetric set \(K\), the pizza quantity of \(K+a\) is polynomial in \(a\) for \(a\) small enough, for example if \(K\) is convex and \(0\in K+a\). We get stronger results in the case of balls, more generally, convex bodies bounded by quadratic hypersurfaces. For example, we prove that the pizza quantity of the ball centered at \(a\) having radius \(R\geq \|a\|\) vanishes for a Coxeter arrangement \(\mathcal{H}\) with \(|\mathcal{H}|-n\) an even positive integer. We also prove the Pizza Theorem for the surface volume: When \(\mathcal{H}\) is a Coxeter arrangement and \(|\mathcal{H}| - n\) is a nonnegative even integer, for an \(n\)-dimensional ball the alternating sum of the \((n-1)\)-dimensional surface volumes of the regions is equal to zero.Topology of symplectomorphism groups and ball-swappingshttps://zbmath.org/1496.530012022-11-17T18:59:28.764376Z"Li, Jun"https://zbmath.org/authors/?q=ai:li.jun.2|li.jun.8|li.jun.3|li.jun.16|li.jun.1|li.jun.7|li.jun.6|li.jun|li.jun.11|li.jun.10"Wu, Weiwei"https://zbmath.org/authors/?q=ai:wu.weiwei|wu.weiwei.1|wu.weiwei.2Summary: In this survey article, we summarize some recent progress and problems on the symplectomorphism groups, with an emphasis on the connection to the space of ball-packings.
For the entire collection see [Zbl 1454.00057].A construction of converging Goldberg-Coxeter subdivisions of a discrete surfacehttps://zbmath.org/1496.530172022-11-17T18:59:28.764376Z"Tao, Chen"https://zbmath.org/authors/?q=ai:tao.chenSummary: It is a fundamental problem to find a good approximating mesh of a given smooth surface and to compare geometries of these two. In the present paper, however, we address this problem from the opposite direction, namely, how we can discover an underlying smooth surface, if it exists, for a given discrete surface (3-valent graph) in \(\mathbb{E}^3\). We construct a method to subdivide a discrete surface in the sense of \textit{M. Kotani} et al. [Comput. Aided Geom. Des. 58, 24--54 (2017; Zbl 1381.65013)] as the solution of the Dirichlet problem for a Goldberg-Coxeter subdivision and prove the sequence of subdivisions forms a Cauchy sequence in the Hausdorif topology. As an application, we give a reason for a carbon network, the Mackay crystal [\textit{A. L. Mackay} and \textit{H. Terrones}, ``Diamond from graphite'', Nature 352, 762 (1991)], to be called a discrete Schwarz P surface (triply periodic minimal surface) known in materials science.The isometric embedding problem for length metric spaceshttps://zbmath.org/1496.530212022-11-17T18:59:28.764376Z"Minemyer, Barry"https://zbmath.org/authors/?q=ai:minemyer.barryIn this paper the author proves a generalization of John Nash's theorem on the isometric embedding of a Riemannian manifold into a Euclidean space. The author proves the following statement:
\textbf{Main Theorem}. Let \(\mathcal{X}\) be a proper \(n\)-dimensional length metric space, and let \(\mathcal{D} \subseteq \mathcal{X}\) be any countable dense subset. Then there exists a collection of geodesics \(\Gamma\) associated to \(\mathcal{D}\) and an embedding \(f :\mathcal{X} \rightarrow \mathbb{R}^{3n+6,1}\) such that \(E(\gamma) = E(f \circ \gamma)\) for all \(\gamma \in \Gamma\). Moreover, if desired, the map \(f\) can be constructed so that its projection onto \(\mathbb{R}^{3n+6,1}\) is not locally Lipschitz. The author explains the proof of the theorem in full detail.
In the introduction the author explains the path to prove the theorem dividing it into 4 majors steps. Step 1 consists of writing the set \(\mathcal{D}\) as the increasing union of finite sets (Section 4). Step 2 (Section 5) consists of the proof of a lemma (Lemma 3) that according to the author \textit{may be of its own independent interest}. In the paper Lemma 3 is the case when \(i=1\) for the proof of the main theorem which is recursive.
\textbf{Lemma 3} Let \((\mathcal{X}, d)\) be an \(n\)-dimensional proper length metric space, let \(\mathcal{D} \subseteq \mathcal{X}\) be any finite subset of \(\mathcal{X}\), and let \(\delta > 0\). Then there exists a map \(f :\mathcal{X} \rightarrow \mathbb{E}^{2n+5}\) which satisfies:
\begin{itemize}
\item[1.] The map \(f\) is \(\delta\)-close to being an embedding. That is, \(f\) satisfies \(f(x) = f(x^{\prime}) \Rightarrow d(x, x^{\prime}) < \delta\)
\item[2.] The map \(f\) is an isometry when restricted to \(\mathcal{D}\). That is, \(d_{f(\mathcal{X})}(f(x), f(x^{\prime})) = d_{\mathcal{X}} (x, x^{\prime})\) for all \(x\), \(x^{\prime}\) \(\in \mathcal{D}\).
\end{itemize}
In Step 3 (Section 6) the author constructs open covers \(\Omega _i\) corresponding to the pair \((\mathcal{D}_i, \Gamma_i)\) in such a way that the mesh of \(\Omega _{i+1}\) is strictly less than one third of the Lebesgue number for \(\Omega _i\) for each \(i\). Step 4 is the conclusion of the proof (Section 7). The article includes three more sections besides the introduction. In Section 2 the energy functional on general metric spaces is studied and how it behaves under perturbations of maps. In Section 3 the theory of indefinite metric polyhedra is discussed and in Section 8 the author presents a ``few of the more `well-known' preliminaries'' and proves that: if \(\mathcal{X}\) is a Finsler manifold which is not Riemannian, and there exists a (path) isometric embedding \(f :\mathcal{X} \rightarrow\mathbb{R}^{p,q}\), \(\pi^+\) and \(\pi^-\) denote the natural projections from \(\mathbb{R}^{p,q}\) onto \(\mathbb{R}^{p,0}\) and \(\mathbb{R}^{0,q}\), respectively, then \(p > 0, q >0\), and both maps \(\pi^+\circ f\) and \(\pi^-\circ f\) are not locally Lipschitz. This proposition is important to understand the main theorem. I would like to call the attention to all the information that the author provides in the introduction, including a long list of remarks related to the main theorem.
Reviewer: Ana Pereira do Vale (Braga)Lightlike and ideal tetrahedrahttps://zbmath.org/1496.530272022-11-17T18:59:28.764376Z"Meusburger, Catherine"https://zbmath.org/authors/?q=ai:meusburger.catherine"Scarinci, Carlos"https://zbmath.org/authors/?q=ai:scarinci.carlosSummary: We give a unified description of tetrahedra with lightlike faces in 3d anti-de Sitter, de Sitter and Minkowski spaces and of their duals in 3d anti-de Sitter, hyperbolic and half-pipe spaces. We show that both types of tetrahedra are determined by a generalized cross-ratio with values in a commutative 2d real algebra that generalizes the complex numbers. Equivalently, tetrahedra with lightlike faces are determined by a pair of edge lengths and their duals by a pair of dihedral angles. We prove that the dual tetrahedra are precisely the generalized ideal tetrahedra introduced by Danciger. Finally, we compute the volumes of both types of tetrahedra as functions of their edge lengths or dihedral angles, obtaining generalizations of the Milnor-Lobachevsky volume formula of ideal hyperbolic tetrahedra.Symplectic homology of convex domains and Clarke's dualityhttps://zbmath.org/1496.530922022-11-17T18:59:28.764376Z"Abbondandolo, Alberto"https://zbmath.org/authors/?q=ai:abbondandolo.alberto"Kang, Jungsoo"https://zbmath.org/authors/?q=ai:kang.jungsooAuthors' abstract: We prove that the Floer complex associated with a convex Hamiltonian function on \(\mathbb{R}^ {2n}\) is isomorphic to the Morse complex of Clarke's dual action functional associated with the Fenchel-dual Hamiltonian. This isomorphism preserves the action filtrations. As a corollary, we obtain that the symplectic capacity from the symplectic homology of a convex domain with smooth boundary coincides with the minimal action of closed characteristics on its boundary.
Reviewer: Alexander Felshtyn (Szczecin)Long time behavior of discrete volume preserving mean curvature flowshttps://zbmath.org/1496.530982022-11-17T18:59:28.764376Z"Morini, Massimiliano"https://zbmath.org/authors/?q=ai:morini.massimiliano"Ponsiglione, Marcello"https://zbmath.org/authors/?q=ai:ponsiglione.marcello"Spadaro, Emanuele"https://zbmath.org/authors/?q=ai:spadaro.emanuele-nunzioSummary: In this paper we analyze the Euler implicit scheme for the volume preserving mean curvature flow. We prove the exponential convergence of the scheme to a finite union of disjoint balls with equal volume for any bounded initial set with finite perimeter.Variational principles and combinatorial \(p\)-th Yamabe flows on surfaceshttps://zbmath.org/1496.531032022-11-17T18:59:28.764376Z"Li, Chunyan"https://zbmath.org/authors/?q=ai:li.chunyan"Lin, Aijin"https://zbmath.org/authors/?q=ai:lin.aijin"Yang, Chang"https://zbmath.org/authors/?q=ai:yang.changThe authors start giving details on PL-metrics (piecewise linear metrics) in Euclidean triangulated manifolds. For these types of metrics, the most natural curvature is the combinatorial Gauss curvature. Considering the combinatorial Gauss-Bonnet formula associated with this curvature one obtains the average of the total combinatorial Gauss curvature designated by \(K_{av}\). The constant combinatorial PL-metric has curvature \(K_{av}\) at all vertices. The Yamabe combinatorial problem deals with the existence of this metric.
The authors start by explaining previous studies on this problem done by other mathematicians. \textit{F. Luo} [Commun. Contemp. Math. 6, No. 5, 765--780 (2004; Zbl 1075.53063)] studied the discrete Yamabe problem through the combinatorial Yamabe flow. Luo showed that the combinatorial Yamabe flow is the negative gradient of a potential functional \(F\). He showed that \(F\) is locally convex and obtained the local rigidity for the curvature map and the local convergence of the combinatorial Yamabe flow. He also conjectured that the combinatorial Yamabe flow converges to a constant curvature PL-metric after a finite number of surgeries on the triangulation. \textit{H. Ge} and \textit{W. Jiang} [Calc. Var. Partial Differ. Equ. 55, No. 6, Paper No. 136, 14 p. (2016; Zbl 1359.53054)] introduced the extended combinatorial Yamabe algorithm to handle possible singularities along the combinatorial Yamabe flow and \textit{X. D. Gu} et al. [J. Differ. Geom. 109, No. 2, 223--256 (2018; Zbl 1396.30008); J. Differ. Geom. 109, No. 3, 431--466 (2018; Zbl 1401.30048)] started doing surgery by ``flipping the algorithm''.
In this paper the authors generalize results for both extended combinatorial Yamabe flow and combinatorial Yamabe flow with surgery. The generalization is done first for \(p>1\) introducing the extended combinatorial \(p\)-th Yamabe flow, which is the extended Yamabe flow when \(p=2\) introduced by Ge and Jiang and then generalize the main results they obtained. They show that the solution to the extended combinatorial \(p\)-th Yamabe flow exists for all time. The details are all explained in Section 2. In Section 3 the authors recall the discrete conformal theory and discrete unifomization theorem established by \textit{X. D. Gu} et al. [J. Differ. Geom. 109, No. 2, 223--256 (2018; Zbl 1396.30008); J. Differ. Geom. 109, No. 3, 431--466 (2018; Zbl 1401.30048)] and generalize their results for the combinatorial \(p\)-th Yamabe flow with surgery. It is shown that for the generalized \(p\)-th flows \(p>1\) and \(p\neq 2\) there exists only curvature convergence but no exponential convergence as in the case of \(p=2\).
Reviewer: Ana Pereira do Vale (Braga)An extended MMP algorithm: wavefront and cut-locus on a convex polyhedronhttps://zbmath.org/1496.650332022-11-17T18:59:28.764376Z"Tateiri, Kazuma"https://zbmath.org/authors/?q=ai:tateiri.kazuma"Ohmoto, Toru"https://zbmath.org/authors/?q=ai:ohmoto.toruA measure of \(Q\)-convexity for shape analysishttps://zbmath.org/1496.683502022-11-17T18:59:28.764376Z"Balázs, Péter"https://zbmath.org/authors/?q=ai:balazs.peter.1|balazs.peter.2"Brunetti, Sara"https://zbmath.org/authors/?q=ai:brunetti.saraSummary: In this paper, we study three basic novel measures of convexity for shape analysis. The convexity considered here is the so-called \(Q\)-convexity, that is, convexity by quadrants. The measures are based on the geometrical properties of \(Q\)-convex shapes and have the following features: (1) their values range from 0 to 1; (2) their values equal 1 if and only if the binary image is \(Q\)-convex; and (3) they are invariant by translation, reflection, and rotation by 90 degrees. We design a new algorithm for the computation of the measures whose time complexity is linear in the size of the binary image representation. We investigate the properties of our measures by solving object ranking problems and give an illustrative example of how these convexity descriptors can be utilized in classification problems.PPLite: zero-overhead encoding of NNC polyhedrahttps://zbmath.org/1496.683512022-11-17T18:59:28.764376Z"Becchi, Anna"https://zbmath.org/authors/?q=ai:becchi.anna"Zaffanella, Enea"https://zbmath.org/authors/?q=ai:zaffanella.eneaSummary: We present an alternative Double Description representation for the domain of NNC (not necessarily closed) polyhedra, together with the corresponding Chernikova-like conversion procedure. The representation uses no slack variable at all and provides a solution to a few technical issues caused by the encoding of an NNC polyhedron as a closed polyhedron in a higher dimension space. We then reconstruct the abstract domain of NNC polyhedra, providing all the operators needed to interface it with commonly available static analysis tools: while doing this, we highlight the efficiency gains enabled by the new representation and we show how the canonicity of the new representation allows for the specification of proper, semantic widening operators. A thorough experimental evaluation shows that our new abstract domain achieves significant efficiency improvements with respect to classical implementations for NNC polyhedra.Analysis of solutions of time-dependent Schrödinger equation of a particle trapped in a spherical boxhttps://zbmath.org/1496.810532022-11-17T18:59:28.764376Z"Nath, Debraj"https://zbmath.org/authors/?q=ai:nath.debraj"Carbó-Dorca, Ramon"https://zbmath.org/authors/?q=ai:carbo-dorca.ramonSummary: Three sets of exact solutions of the time-dependent Schrödinger equation of a particle that is trapped in a spherical box with a moving boundary wall have been calculated analytically. For these solutions, some physical quantities such as time-dependent average energy, average force, disequilibrium, quantum similarity measures as well as quantum similarity index have been investigated. Moreover, these solutions are compared concerning these physical quantities. The time-correlation functions among these solutions are investigated.A convex analysis view of the barrier problemhttps://zbmath.org/1496.901112022-11-17T18:59:28.764376Z"Bessenyei, Mihály"https://zbmath.org/authors/?q=ai:bessenyei.mihaly"Tóth, Norbert"https://zbmath.org/authors/?q=ai:toth.norbertSummary: Besides the simplex algorithm, linear programs can also be solved via interior point methods. The theoretical background of such algorithms is the classical log-barrier problem. The aim of this note is to study and generalize the barrier problem using the standard tools of Convex Analysis.