Recent zbMATH articles in MSC 12K10, 14T, 52https://zbmath.org/atom/cc/52,14T,12K102024-04-02T17:33:48.828767ZWerkzeugTwenty years of progress of JCDCG\(^3\)https://zbmath.org/1529.010182024-04-02T17:33:48.828767Z"Akiyama, Jin"https://zbmath.org/authors/?q=ai:akiyama.jin"Ito, Hiro"https://zbmath.org/authors/?q=ai:ito.hiro"Sakai, Toshinori"https://zbmath.org/authors/?q=ai:sakai.toshinori"Uno, Yushi"https://zbmath.org/authors/?q=ai:uno.yushiSummary: The conferences in the series of the Japan Conference on Discrete and Computational Geometry, Graphs and Games (JCDCG\(^3\)) have been held annually since 1997, except for 2008. Since 1997, almost one thousand research results have been presented in total at the conferences, and 11 post-conference proceedings and 6 special issues of journals have been published. To celebrate the 20th Anniversary of JCDCG\(^3\), a summary of the notable results published in those proceedings are presented in this article. We focus on six areas such as games and puzzles, dissection and reversibility, foldings and unfoldings, point sets, visibility, and geometric and topological graph theory.A Catalanization map on the symmetric grouphttps://zbmath.org/1529.050122024-04-02T17:33:48.828767Z"Can, Mahir Bilen"https://zbmath.org/authors/?q=ai:can.mahir-bilen"Nelson, Luke"https://zbmath.org/authors/?q=ai:nelson.luke"Treat, Kevin"https://zbmath.org/authors/?q=ai:treat.kevinSummary: In this article we investigate a self-map of the symmetric group. We show that the set of plus-indecomposable permutations is stable under our map. Furthermore, the fixed points are precisely the 231avoiding permutations. Finally, by using our map we obtain a nested sequence of polytopes that starts with the permutahedron and ends with the associahedron.The extremals of the Alexandrov-Fenchel inequality for convex polytopeshttps://zbmath.org/1529.050322024-04-02T17:33:48.828767Z"Shenfeld, Yair"https://zbmath.org/authors/?q=ai:shenfeld.yair"van Handel, Ramon"https://zbmath.org/authors/?q=ai:van-handel.ramonSummary: The Alexandrov-Fenchel inequality, a far-reaching generalization of the classical isoperimetric inequality to arbitrary mixed volumes, lies at the heart of convex geometry. The characterization of its extremal bodies is a long-standing open problem that dates back to \textit{A. D. Aleksandrov}'s original paper [Rec. Math. Moscou, n. Ser. 3, 27--46 (1938; Zbl 0018.42402)]. The known extremals already form a very rich family, and even the fundamental conjectures on their general structure, due to \textit{R. Schneider} [in: Discrete geometry and convexity, Proc. Conf., New York 1982, Ann. N.Y. Acad. Sci. 440, 132--141 (1985; Zbl 0567.52004)], are incomplete. In this paper, we completely settle the extremals of the Alexandrov-Fenchel inequality for convex polytopes. In particular, we show that the extremals arise from the combination of three distinct mechanisms: translation, support, and dimensionality. The characterization of these mechanisms requires the development of a diverse range of techniques that shed new light on the geometry of mixed volumes of non-smooth convex bodies. Our main result extends further beyond polytopes in a number of ways, including to the setting of quermassintegrals of arbitrary convex bodies. As an application of our main result, we settle a question of \textit{R. P. Stanley} [J. Comb. Theory, Ser. A 31, 56--65 (1981; Zbl 0484.05012)] on the extremal behavior of certain \(\log\)-concave sequences that arise in the combinatorics of partially ordered sets.Nets in \(\mathbb{P}^2\) and Alexander dualityhttps://zbmath.org/1529.050332024-04-02T17:33:48.828767Z"Abdallah, Nancy"https://zbmath.org/authors/?q=ai:abdallah.nancy"Schenck, Hal"https://zbmath.org/authors/?q=ai:schenck.halSummary: A net in \(\mathbb{P}^2\) is a configuration of lines \(\mathcal{A}\) and points \(X\) satisfying certain incidence properties. Nets appear in a variety of settings, ranging from quasigroups to combinatorial design to classification of Kac-Moody algebras to cohomology jump loci of hyperplane arrangements. For a matroid \(M\) and rank \(r\), we associate a monomial ideal (a monomial variant of the Orlik-Solomon ideal) to the set of flats of \(M\) of rank \(\le r\). In the context of line arrangements in \(\mathbb{P}^2\), applying Alexander duality to the resulting ideal yields insight into the combinatorial structure of nets.Flag matroids with coefficientshttps://zbmath.org/1529.050362024-04-02T17:33:48.828767Z"Jarra, Manoel"https://zbmath.org/authors/?q=ai:jarra.manoel"Lorscheid, Oliver"https://zbmath.org/authors/?q=ai:lorscheid.oliverSummary: This paper is a direct generalization of Baker-Bowler theory to flag matroids [\textit{M. Baker} and \textit{N. Bowler}, Adv. Math. 343, 821--863 (2019; Zbl 1404.05022)], including its moduli interpretation as developed by \textit{M. Baker} and \textit{O. Lorscheid} [ibid. 390, Article ID 107883, 118 p. (2021; Zbl 1479.05045)] for matroids. More explicitly, we extend the notion of flag matroids to flag matroids over any tract, provide cryptomorphic descriptions in terms of basis axioms (Grassmann-Plücker functions), circuit/vector axioms and dual pairs, including additional characterizations in the case of perfect tracts. We establish duality of flag matroids and construct minors. Based on the theory of ordered blue schemes, we introduce flag matroid bundles and construct their moduli space, which leads to algebro-geometric descriptions of duality and minors. Taking rational points recovers flag varieties in several geometric contexts: over (topological) fields, in tropical geometry, and as a generalization of the MacPhersonian.Minimal tropical bases for Bergman fans of matroidshttps://zbmath.org/1529.050372024-04-02T17:33:48.828767Z"Nakajima, Yasuhito"https://zbmath.org/authors/?q=ai:nakajima.yasuhitoSummary: The Bergman fan of a matroid is the intersection of tropical hyperplanes defined by the circuits. A tropical basis is a subset of the circuit set that defines the Bergman fan. \textit{J. Yu} and \textit{D. S. Yuster} [``Representing tropical linear spaces by circuits'', Preprint, \url{arXiv:math/0611579}] posed a question whether every simple regular matroid has a unique minimal tropical basis of its Bergman fan, and verified it for graphic, cographic matroids and \(R_{10}\). We show every simple binary matroid has a unique minimal tropical basis. Since the regular matroid is binary, we positively answered the question.On the monophonic convexity in complementary prismshttps://zbmath.org/1529.050522024-04-02T17:33:48.828767Z"P. K., Neethu"https://zbmath.org/authors/?q=ai:p-k.neethu"Chandran S. V., Ullas"https://zbmath.org/authors/?q=ai:chandran-s-v.ullas"Nascimento, Julliano R."https://zbmath.org/authors/?q=ai:nascimento.julliano-rosaSummary: A set \(S\) of vertices of a graph \(G\) is monophonic convex if \(S\) contains all the vertices belonging to any induced path connecting two vertices of \(S\). The cardinality of a maximum proper monophonically convex set of \(G\) is called the monophonic convexity number of \(G\). The monophonic interval of a set \(S\) of vertices of \(G\) is the set \(S\) together with every vertex belonging to any induced path connecting two vertices of \(S\). The cardinality of a minimum set \(S \subseteq V(G)\) whose monophonic interval is \(V(G)\) is called the monophonic interval number of \(G\). The monophonic convex hull of a set \(S\) of vertices of \(G\) is the smallest monophonically convex set containing \(S\) in \(G\). The cardinality of a minimum set \(S \subseteq V(G)\) whose monophonic convex hull is \(V(G)\) is called the monophonic hull number of \(G\). The complementary prism \(G \overline{G}\) of \(G\) is obtained from the disjoint union of \(G\) and its complement \(\overline{G}\) by adding the edges of a perfect matching between them. In this work, we show that the corresponding decision problem to the monophonic convexity number is NP-complete even for complementary prisms. On the other hand, we show that the monophonic interval number and the monophonic hull number can be determined in linear time for the complementary prisms of all graphs.\(l_1\)-embeddability of shifted quadrilateral cylinder graphshttps://zbmath.org/1529.051102024-04-02T17:33:48.828767Z"Wang, Guangfu"https://zbmath.org/authors/?q=ai:wang.guangfu"Xiong, Zhikun"https://zbmath.org/authors/?q=ai:xiong.zhikun"Chen, Lijun"https://zbmath.org/authors/?q=ai:chen.lijun.1|chen.lijunSummary: A connected graph \(G\) is called \(l_1\)-embeddable, if it can be isometrically embedded into the \(l_1\)-space. The shifted quadrilateral cylinder graph \(O_{m,n,k}\) is a class of quadrilateral tilings on a cylinder obtained by rolling the grid graph \(P_m \square P_n\) via shifting \(k\) positions. In this article, we determine that all the \(O_{m,n,k}\) are not \(l_1\)-embeddable except for \(O_{m,n,0}\) and \(O_{m,3,1}\).Empty triangles in generalized twisted drawings of \(K_n\)https://zbmath.org/1529.051162024-04-02T17:33:48.828767Z"García, Alfredo"https://zbmath.org/authors/?q=ai:garcia.alfredo-daniel"Tejel, Javier"https://zbmath.org/authors/?q=ai:tejel.javier"Vogtenhuber, Birgit"https://zbmath.org/authors/?q=ai:vogtenhuber.birgit"Weinberger, Alexandra"https://zbmath.org/authors/?q=ai:weinberger.alexandraSummary: Simple drawings are drawings of graphs in the plane such that vertices are distinct points, edges are Jordan arcs connecting their endpoints, and edges intersect at most once (either in a proper crossing or in a shared endpoint). Simple drawings are generalized twisted if there is a point \(O\) such that every ray emanating from \(O\) crosses every edge of the drawing at most once, and there is a ray emanating from \(O\) which crosses every edge exactly once. We show that all generalized twisted drawings of \(K_n\) contain exactly \(2n-4\) empty triangles, by this making a substantial step towards proving the conjecture that any simple drawing of \(K_n\) contains at least \(2n-4\) empty triangles.The \(m=2\) amplituhedron and the hypersimplexhttps://zbmath.org/1529.051622024-04-02T17:33:48.828767Z"Parisi, Matteo"https://zbmath.org/authors/?q=ai:parisi.matteo"Sherman-Bennett, Melissa"https://zbmath.org/authors/?q=ai:sherman-bennett.melissa-u"Williams, Lauren"https://zbmath.org/authors/?q=ai:williams.lauren-kSummary: The hypersimplex \(\Delta_{k+1},n\) is the image of the positive Grassmannian \(Gr^{\geq 0}_{k+1,n}\) under the moment map. It is a polytope of dimension \(n-1\) in \(\mathbb{R}^n\). Meanwhile, the amplituhedron \(\mathcal{A}^Z_{n,k,2}\) is the image of \(Gr^{\geq 0}_{k,n}\) under an amplituhedron map \(\widetilde{Z}\) induced by a positive matrix \(Z\). Introduced in the context of scattering amplitudes, it is not a polytope, and is a full dimensional subset of \(Gr_{k,k+2}\). Nevertheless, there seem to be remarkable connections between these two objects, as conjectured by Lukowski-Parisi-Williams (LPW) [\textit{T. Łukowski} et al., Int. Math. Res. Not. 2023, No. 3 (2023; \url{doi:10.1093/imrn/rnad010})]. We use ideas from oriented matroid theory, total positivity, and the geometry of the hypersimplex and positroid polytopes to obtain a deeper understanding of the amplituhedron. We show that the inequalities cutting out positroid polytopes -- moment map images of positroid cells -- translate into sign conditions cutting out Grasstopes -- amplituhedron map images of positroid cells. Moreover, we subdivide the amplituhedron into chambers, just as the hypersimplex can be subdivided into simplices -- with both chambers and simplices enumerated by the Eulerian numbers. We use these properties to prove the main conjecture of (LPW): a collection of positroid polytopes is a tiling of the hypersimplex if and only if the collection of T-dual Grasstopes is a tiling of the amplituhedron \(\mathcal{A}^Z_{n,k,2}\) for all \(Z\). We also prove Arkani-Hamed-Thomas-Trnka's conjectural sign-flip characterization of \(\mathcal{A}^Z_{n,k,2}\) [\textit{N. Arkani-Hamed} et al., J. High Energy Phys. 2018, No. 1, Paper No. 16, 41 p. (2018; Zbl 1384.81130)].On higher dimensional arithmetic progressions in Meyer setshttps://zbmath.org/1529.110152024-04-02T17:33:48.828767Z"Klick, Anna"https://zbmath.org/authors/?q=ai:klick.anna"Strungaru, Nicolae"https://zbmath.org/authors/?q=ai:strungaru.nicolaeSummary: In this paper we study the existence of higher dimensional arithmetic progressions in Meyer sets. We show that the case when the ratios are linearly dependent over \(\mathbb{Z}\) is trivial and focus on arithmetic progressions for which the ratios are linearly independent. Given a Meyer set \(\Lambda\) and a fully Euclidean model set \(\varLambda(W)\) with the property that finitely many translates of \(\varLambda(W)\) cover \(\Lambda \), we prove that we can find higher dimensional arithmetic progressions of arbitrary length with \(k\) linearly independent ratios in \(\Lambda\) if and only if \(k\) is at most the rank of the \({\mathbb Z}\)-module generated by \((\varLambda(W))\). We use this result to characterize the Meyer sets that are subsets of fully Euclidean model sets.Piercing the chessboardhttps://zbmath.org/1529.110862024-04-02T17:33:48.828767Z"Ambrus, Gergely"https://zbmath.org/authors/?q=ai:ambrus.gergely"Bárány, Imre"https://zbmath.org/authors/?q=ai:barany.imre"Frankl, Péter"https://zbmath.org/authors/?q=ai:frankl.peter"Varga, Dániel"https://zbmath.org/authors/?q=ai:varga.danielLet \(h_n\) (respectively \(p_n\)) denote the minimum number of lines needed to intersect (respectively pierce) all the cells of an \(n \times n\) chessboard. The paper proves that \(h_n = \lceil \frac{n}{2} \rceil \) for \(n\geq 1\). For \(p_n\) the authors give the bounds \(0.7n \leq p_n \leq n-1\) (for \(n\geq 3\)) where they conjecture the upper bound to be sharp. To prove the lower bound the authors use linear programming methods and demonstrate their limitations.
Reviewer: Gabriele Nebe (Aachen)Double Schubert polynomials do have saturated Newton polytopeshttps://zbmath.org/1529.140342024-04-02T17:33:48.828767Z"Castillo, Federico"https://zbmath.org/authors/?q=ai:castillo.federico"Cid-Ruiz, Yairon"https://zbmath.org/authors/?q=ai:cid-ruiz.yairon"Mohammadi, Fatemeh"https://zbmath.org/authors/?q=ai:mohammadi.fatemeh"Montaño, Jonathan"https://zbmath.org/authors/?q=ai:montano.jonathanSummary: We prove that double Schubert polynomials have the saturated Newton polytope property. This settles a conjecture by Monical, Tokcan and Yong. Our ideas are motivated by the theory of multidegrees. We introduce a notion of standardization of ideals that enables us to study nonstandard multigradings. This allows us to show that the support of the multidegree polynomial of each Cohen-Macaulay prime ideal in a nonstandard multigrading, and in particular, that of each Schubert determinantal ideal is a discrete polymatroid.Tropical geometry forwards and backwardshttps://zbmath.org/1529.140392024-04-02T17:33:48.828767Z"Ranganathan, Dhruv"https://zbmath.org/authors/?q=ai:ranganathan.dhruvTropical geometry is in some sense a combinatorial analogue of algebraic geometry and the two theories are linked by the \emph{tropicalization} process, which turns an algebraic variety into a discrete, piecewise linear, tropical object. Unfortunately, this process is almost never invertible, which leads to the \emph{tropical inverse} or \emph{realizability problem}.
In in the present survey article, the author presents successful applications of tropical geometry to algebro-geometric problems (Brill-Noether theory, enumerative geometry), but also illustrates the intricacies of the realizability problem.
Reviewer: Felix Röhrle (Frankfurt am Main)A greedy algorithm to compute arrangements of lines in the projective planehttps://zbmath.org/1529.200612024-04-02T17:33:48.828767Z"Cuntz, Michael"https://zbmath.org/authors/?q=ai:cuntz.michaelSummary: We introduce a greedy algorithm optimizing arrangements of lines with respect to a property. We apply this algorithm to the case of simpliciality: it recovers all known simplicial arrangements of lines in a very short time and also produces a yet unknown simplicial arrangement with 35 lines. We compute a (certainly incomplete) database of combinatorially simplicial complex arrangements of hyperplanes with up to 50 lines. Surprisingly, it contains several examples whose matroids have an infinite space of realizations up to projectivities.Complete conformal metrics of negative Ricci curvature in open manifoldshttps://zbmath.org/1529.352762024-04-02T17:33:48.828767Z"Chen, Li"https://zbmath.org/authors/?q=ai:chen.li.6|chen.li.19|chen.li|chen.li.44|chen.li.62|chen.li.10|chen.li.47|chen.li.1|chen.li.14|chen.li.9|chen.li.43|chen.li.16|chen.li.45|chen.li.20|chen.li.5"He, Yan"https://zbmath.org/authors/?q=ai:he.yan.1|he.yan"Li, Mingming"https://zbmath.org/authors/?q=ai:li.mingming"Tu, Chengming"https://zbmath.org/authors/?q=ai:tu.chengmingSummary: In this paper, we study the problem of finding complete conformal metrics determined by some symmetric function of the modified Schouten tensor in open manifolds. We prove the existence of such metrics by solving a class of fully nonlinear elliptic equations with the infinity Dirichlet boundary condition.The covariogram problemhttps://zbmath.org/1529.420102024-04-02T17:33:48.828767Z"Bianchi, Gabriele"https://zbmath.org/authors/?q=ai:bianchi.gabrieleSummary: The covariogram \(g_X\) of a measurable set \(X\) in \(\mathbb{R}^n\) is the function which associates to each \(x \in \mathbb{R}^n\) the measure of the intersection of \(X\) with \(X + x\). We are interested in understanding what information about a set can be obtained from its covariogram. \textit{G. Matheron} [Le covariogramme géometrique des compacts convexes de \(\mathbb R^2\). Techn. Rep., Mines ParisTech (1986), \url{http://cg.ensmp.fr/bibliotheque/public/MATHERON_Rapport_00258.pdf}] asked whether a convex body \(K\) is determined from the knowledge of \(g_K\), and this is known as the covariogram problem.
The covariogram appears in very different contexts, and the covariogram problem can be rephrased in different terms. For instance, it is equivalent to determining the characteristic function \(1_K\) of \(K\) from the modulus of its Fourier transform \(\widehat{1_K}\) in \(\mathbb{R}^n\), a particular instance of the phase retrieval problem. The covariogram problem has also a discrete counterpart.
We survey the known results and the methods.
For the entire collection see [Zbl 1519.42002].Spectral sets and weak tilinghttps://zbmath.org/1529.420122024-04-02T17:33:48.828767Z"Kolountzakis, Mihail N."https://zbmath.org/authors/?q=ai:kolountzakis.mihail-n"Lev, Nir"https://zbmath.org/authors/?q=ai:lev.nir"Matolcsi, Máté"https://zbmath.org/authors/?q=ai:matolcsi.mateA set \(\Omega\in\mathbb R^d\) is said to be spectral if the space \(L^2(\Omega)\) admits an orthogonal basis of exponential functions. \textit{B. Fuglede} [J. Funct. Anal. 16, 101--121 (1974; Zbl 0279.47014)] conjectured that \(\Omega\) is spectral if and only if it can tile the space by translations. While this conjecture was disproved for general sets, it was recently proved that the Fuglede conjecture does hold for the class of convex bodies in \(\mathbb R^d\). The proof was based on a new geometric necessary condition for spectrality, called ``weak tiling''. In this paper, further properties of the weak tiling notion are studied, and applications to convex bodies, non-convex polytopes, product domains and Cantor sets of positive measure are given.
Reviewer: Elijah Liflyand (Ramat-Gan)A topological characterisation of Haar null convex setshttps://zbmath.org/1529.460312024-04-02T17:33:48.828767Z"Ravasini, Davide"https://zbmath.org/authors/?q=ai:ravasini.davideThis paper is the third in the author's series of new results concerning Haar positivity for closed, convex sets \(C\) in separable Banach spaces \(X\).
If we combine results from the first two, we find the following characterizations: ``\(C\subset X\) is Haar positive if and only if it is not Haar meager if and only if there exists \(r>0\) such that for any compact \(K\subseteq rB_X\) there is \(x\in X\) such that \(x+K\subseteq C\)'' [\textit{D.~Ravasini}, Bull. Lond. Math. Soc. 55, No.~1, 137--148 (2023; Zbl 1527.46015)]. Further, the latter property turns out to be an equivalent formulation of compactivorousity in general in Banach spaces [\textit{D.~Ravasini}, Proc. Am. Math. Soc. 150, No.~5, 2121--2129 (2022; Zbl 1493.46037)].
It is classical that a closed, convex subset of \(\mathbb{R}^n\) is Lebesgue positive if and only if it has non-empty interior. This was generalized by \textit{E.~Matoušková} [Bull. Lond. Math. Soc. 33, No.~6, 711--714 (2001; Zbl 1033.46018)]: A closed, convex subset of a separable reflexive space is Haar positive if and only if it has non-empty interior in the norm topology. Using the above mentioned new formulations of Haar positivity, the author in [Bull. Lond. Math. Soc. 55, No.~1, 137--148 (2023; Zbl 1527.46015)] generalized Matoušková's result: A weak-star closed, convex subset of a separable dual space is Haar positive if and only if it has non-empty interior in the norm topology.
Using quantification techniques of the compactivorousity version of Haar positivity above in general Banach spaces, the author in the present paper obtains as a special case that could be the ultimate generalization: ``A closed, convex subset of a separable Banach space is Haar positive if and only if its weak star closure in \(X^{\ast\ast}\) has non-empty interior in the norm topology of \(X^{\ast\ast}\).''
Let us return to general Banach spaces and let \(E^{(1)}\) denote the first derived set of \(E\subset X^\ast\). It is clear that \(E^{(1)}\subseteq \overline{E}^{w^\ast}\) and well known that the inclusion may be proper. The author obtains the following: ``For a closed, convex set \(C\) in a Banach space \(X\), \(C^{(1)}\) (taken in \(X^{\ast\ast}\)) cannot have empty norm-interior unless \(\overline{C}^{w^\ast}\) has.''
In a final section the author studies the complete metric space \(\mathcal{C}(X)\) whose elements are the closed, convex, bounded subsets of the Banach space \(X\) and the metric is the Hausdorff distance. General results on some quantifiers are derived and as a special result he obtains that for separable \(X\), the family of Haar positive elements of \(\mathcal{C}(X)\) forms an open subset. So, ``a convergent sequence of Haar-null sets in \(\mathcal{C}(X)\) has Haar-null limit.''
Reviewer: Olav Nygaard (Kristiansand)Composition operators, convexity of their Berezin range and related questionshttps://zbmath.org/1529.470452024-04-02T17:33:48.828767Z"Augustine, Athul"https://zbmath.org/authors/?q=ai:augustine.athul"Garayev, M."https://zbmath.org/authors/?q=ai:garaev.moubariz-z|garayev.mubariz-tapdigoglu"Shankar, P."https://zbmath.org/authors/?q=ai:shankar.p-mohana|shankar.priti|shankar.p-nSummary: The Berezin range of a bounded operator \(T\) acting on a reproducing kernel Hilbert space \(\mathcal{H}\) is the set \(\mathrm{Ber}(T) := \{\langle T\hat{k}_x, \hat{k}_x\rangle_{\mathcal{H}}: x\in X\}\), where \(\hat{k}_x\) is the normalized reproducing kernel for \(\mathcal{H}\) at \(x\in X\). In general, the Berezin range of an operator is not convex. In this paper, we discuss the convexity of range of the Berezin transforms. We characterize the convexity of the Berezin range for a class of composition operators acting on the Hardy space and the Bergman space of the unit disk. Also for so-called superquadratic functions, we prove the Berezin set mapping theorem for positive self-adjoint operators \(A\) on the reproducing kernel Hilbert space \(\mathcal{H}(\Omega)\), namely we prove that \(f(\mathrm{Ber}(\Phi(A))) = \mathrm{Ber}(\Phi(f(A)))\), where \(\Phi: \mathcal{B}(\mathcal{H}(\Omega))\rightarrow\mathcal{B}(\mathcal{K}(Q))\) is a normalized positive linear map.Remarks on the KKM theory of abstract convex minimal spaceshttps://zbmath.org/1529.470882024-04-02T17:33:48.828767Z"Park, Sehie"https://zbmath.org/authors/?q=ai:park.sehieSummary: Recently, \textit{M. Alimohammady} et al. [Nonlinear Funct. Anal. Appl. 13, No. 3, 483--492 (2008; Zbl 1167.26011); Nonlinear Anal., Model. Control 10, No. 4, 305--314 (2005; Zbl 1147.54323); Nonlinear Funct. Anal. Appl. 13, No. 4, 597--611 (2008; Zbl 1165.26005)], \textit{R. Darzi} et al. [Filomat 25, No. 4, 165--176 (2011; Zbl 1265.26028)], and \textit{M. R. Delavar} et al. [Nonlinear Funct. Anal. Appl. 16, No. 2, 201--210 (2011; Zbl 1258.49003)] dealt with some results in the KKM theory on generalized convex minimal spaces. By establishing a kind of the KKM principle in these spaces, they obtained some results on coincidence or fixed point theorems and others. Our aim in the present paper is to show that their results are consequences of corresponding ones for abstract convex minimal spaces in our previous paper [\textit{S. Park}, Nonlinear Funct. Anal. Appl. 13, No. 2, 179--191 (2008; Zbl 1511.54043)] and hence, can be extended to a more general setting.Review of recent studies on the KKM theory. IIhttps://zbmath.org/1529.470892024-04-02T17:33:48.828767Z"Park, Sehie"https://zbmath.org/authors/?q=ai:park.sehieSummary: In our previous survey [\textit{S. Park}, Nonlinear Anal. Forum 15, 1--12 (2010; Zbl 1307.47062)], we gave a short history of the KKM theory and reviewed its current study by recalling our previous comments or surveys in a sequence of papers. The present survey is a continuation of [loc.\,cit.]\ and to review some recent works on the theory mainly due to other authors. On this occasion, we give some corrections on [\textit{S. Park}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 2, 549--554 (2010; Zbl 1183.52001)].The Douglas lemma for von Neumann algebras and some applicationshttps://zbmath.org/1529.471302024-04-02T17:33:48.828767Z"Nayak, Soumyashant"https://zbmath.org/authors/?q=ai:nayak.soumyashantIn the present paper, we find a discussion of some applications of the Douglas factorization lemma in the context of von Neumann algebras. The author gives a constructive proof of this lemma and some new results about left ideals of von Neumann algebras.
Let \(\mathcal{H}\)\ be a complex Hilbert space, \(\mathcal{B}(\mathcal{H})\) the set of bounded operators on \(\mathcal{H}\) and \(\mathcal{R}\) be a von Neumann algebra acting on \(\mathcal{H}\). It shown that every left ideal of \(\mathcal{R}\) can be realized as the intersection of a left ideal of \( \mathcal{B}(\mathcal{H})\) with \(\mathcal{R}\). Also, the author generalizes a result by Loebl and Paulsen [\textit{R. I. Loebl} and \textit{V. I. Paulsen}, Linear Algebra Appl. 35, 63--78 (1981; Zbl 0448.46038)] pertaining to \(C^{\ast }\)-convex subsets of \( \mathcal{B}(\mathcal{H})\) to the context of \(\mathcal{R}\)-bimodules.
Reviewer: Elhadj Dahia (Bou Saâda)Symplectic 4-dimensional semifields of order \(8^4\) and \(9^4\)https://zbmath.org/1529.510042024-04-02T17:33:48.828767Z"Lavrauw, Michel"https://zbmath.org/authors/?q=ai:lavrauw.michel"Sheekey, John"https://zbmath.org/authors/?q=ai:sheekey.johnThe authors classify symplectic 4-dimensional semifields over \({\mathbb F}_q\), for \(q \le 9\). They show that every symplectic 4-dimensional semifield over \({\mathbb F}_q\) for \(q\) even, \(q \le 8\) is associative and hence a field. Every symplectic 4-dimensional semifield over \({\mathbb F}_q\) for \(q\) odd, \(q \le 9\) is Knuth-equivalent to a Dickson commutative semifield. For \(q \le 7\), these results were known previously and are confirmed by the authors.
The proofs are mainly computational and the authors carefully describe their ideas and algorithms.
Reviewer: Norbert Knarr (Stuttgart)Some results about equichordal convex bodieshttps://zbmath.org/1529.520012024-04-02T17:33:48.828767Z"Jerónimo-Castro, Jesús"https://zbmath.org/authors/?q=ai:jeronimo-castro.jesus"Jimenez-Lopez, Francisco G."https://zbmath.org/authors/?q=ai:jimenez-lopez.francisco-g"Morales-Amaya, Efrén"https://zbmath.org/authors/?q=ai:morales-amaya.efrenSummary: Let \(K\) and \(L\) be two convex bodies in \({\mathbb{R}}^n\), \(n\ge 2\), with \(L\subset{{\,\operatorname{int}\,}}K\). We say that \(L\) is an \textit{equichordal body} for \(K\) if every chord of \(K\) tangent to \(L\) has length equal to a given fixed value \(\lambda \). \textit{J. A. Barker} and \textit{D. G. Larman} [Discrete Math. 241, No. 1--3, 79--96 (2001; Zbl 0999.52002)] proved that if \(L\) is a ball, then \(K\) is a ball concentric with \(L\). In this paper we prove that there exist an infinite number of closed curves, different from circles, which possess an equichordal convex body. If the dimension of the space is more than or equal to 3, then only Euclidean balls possess an equichordal convex body.Diversities and the generalized circumradiushttps://zbmath.org/1529.520022024-04-02T17:33:48.828767Z"Bryant, David"https://zbmath.org/authors/?q=ai:bryant.david"Huber, Katharina T."https://zbmath.org/authors/?q=ai:huber.katharina-t"Moulton, Vincent"https://zbmath.org/authors/?q=ai:moulton.vincent-l"Tupper, Paul F."https://zbmath.org/authors/?q=ai:tupper.paul-fThe authors continue the study of the so-called diversities that were introduced in [\textit{D. Bryant} and \textit{P. F. Tupper}, Adv. Math. 231, No. 6, 3172--3198 (2012; Zbl 1256.54055); Discrete Math. Theor. Comput. Sci. 16, No. 2, 1--20 (2014; Zbl 1294.05117)].
They introduce the generalized circumradius of a subset of \({\mathbb R}^d\) by replacing the euclidean ball with an arbitrary convex body called a kernel and prove that the generalized circumradius is a diversity. Also they characterize Minkowski diversities and elaborate the cases that the kernel is a simplex or parallelotope.
Reviewer: S. S. Kutateladze (Novosibirsk)Elliptic polytopes and invariant norms of linear operatorshttps://zbmath.org/1529.520032024-04-02T17:33:48.828767Z"Mejstrik, Thomas"https://zbmath.org/authors/?q=ai:mejstrik.thomas"Protasov, Valdimir Yu."https://zbmath.org/authors/?q=ai:protasov.valdimir-yuSummary: Elliptic polytopes are convex hulls of several concentric plane ellipses in \(\mathbb{R}^d\). They arise in applications as natural generalizations of usual polytopes. In particular, they define invariant convex bodies of linear operators, optimal Lyapunov norms for linear dynamical systems, etc. To construct elliptic polytopes one needs to decide whether a given ellipse is contained in the convex hull of other ellipses. We analyse the computational complexity of this problem and show that for \(d=2, 3\), it admits an explicit solution. For larger \(d\), two geometric methods for approximate solution are presented. Both use the convex optimization tools. The efficiency of the methods is demonstrated in two applications: the construction of extremal norms of linear operators and the computation of the joint spectral radius/Lyapunov exponent of a family of matrices.Facets of spherical random polytopeshttps://zbmath.org/1529.520042024-04-02T17:33:48.828767Z"Bonnet, Gilles"https://zbmath.org/authors/?q=ai:bonnet.gilles"O'Reilly, Eliza"https://zbmath.org/authors/?q=ai:oreilly.elizaLet \(X_1, \dots, X_n\) be i.i.d. unit vectors (random points) uniformly distributed on the sphere \(\mathbb{S}^{d-1}\) (unit sphere centered at the origin), \(n > d\ge 2\), and denote by \(P_{n, d}\) the convex hull of these points. A facet of \(P_{n, d}\) has height \(h\in [ -1, 1]\) if its supporting hyperplane has the form \(\{x\in\mathbb{R}^d: \langle x, u\rangle = h\}\) for some unit vector \(u\in\mathbb{S}^{d-1}\) and the polytope is contained in the half space \(\{x\in\mathbb{R}^d: \langle x, u\rangle\le h\}\). Note that a facet can have a negative height. In fact, a polytope contains the origin in its interior precisely when all facets have positive height.
The paper investigates the heights of the facets of \(P_{n, d}\) as \(n\to \infty\) and \(d\) is either constant or tends to \(\infty\). There exists an extensive literature on the properties of these polytopes, see [\textit{R. Schneider} and \textit{W. Weil}, Stochastic and integral geometry. Berlin: Springer (2008; Zbl 1175.60003)] and [\textit{D. Hug}, Lect. Notes Math. 2068, 205--238 (2013; Zbl 1275.60017)]. The organization of the paper is as follows. Sections 2--4 present the main results of the paper. Section 5 describes related results from the literature in fixed dimension, and describes how the main results of the paper extend these formulas to the case when \(d\) tends to infinity. Finally, Section 6 presents the proofs in increasing order of the asymptotic regimes for \(n\).
Reviewer: Viktor Ohanyan (Erevan)Quantitative Helly-type theorems via sparse approximationhttps://zbmath.org/1529.520052024-04-02T17:33:48.828767Z"Almendra-Hernández, Víctor Hugo"https://zbmath.org/authors/?q=ai:almendra-hernandez.victor-hugo"Ambrus, Gergely"https://zbmath.org/authors/?q=ai:ambrus.gergely"Kendall, Matthew"https://zbmath.org/authors/?q=ai:kendall.matthewSummary: We prove the following sparse approximation result for polytopes. Assume that \(Q\) is a polytope in John's position. Then there exist at most \(2d\) vertices of \(Q\) whose convex hull \(Q'\) satisfies \(Q \subseteq - 2d^2\,Q'\). As a consequence, we retrieve the best bound for the quantitative Helly-type result for the volume, achieved by Brazitikos, and improve on the strongest bound for the quantitative Helly-type theorem for the diameter, shown by Ivanov and Naszódi: We prove that given a finite family \(\mathcal{F}\) of convex bodies in \(\mathbb{R}^d\) with intersection \(K\), we may select at most \(2d\) members of \(\mathcal{F}\) such that their intersection has volume at most \((c d)^{3d /2} {\,{\text{vol}}\,}K\), and it has diameter at most \(2d^2 {{\,\operatorname{diam}\,}}K\), for some absolute constant \(c>0\).Fractional Helly theorem for Cartesian products of convex setshttps://zbmath.org/1529.520062024-04-02T17:33:48.828767Z"Chakraborti, Debsoumya"https://zbmath.org/authors/?q=ai:chakraborti.debsoumya"Kim, Jaehoon"https://zbmath.org/authors/?q=ai:kim.jaehoon"Kim, Jinha"https://zbmath.org/authors/?q=ai:kim.jinha"Kim, Minki"https://zbmath.org/authors/?q=ai:kim.minki"Liu, Hong"https://zbmath.org/authors/?q=ai:liu.hong.1Summary: Helly's theorem and its variants show that for a family of convex sets in Euclidean space, local intersection patterns influence global intersection patterns. A classical result of Eckhoff in 1988 provided an optimal fractional Helly theorem for axis-aligned boxes, which are Cartesian products of line segments. Answering a question raised by Bárány and Kalai, and independently Lew, we generalize Eckhoff's result to Cartesian products of convex sets in all dimensions. In particular, we prove that given \(\alpha \in (1-1/t^d,1]\) and a finite family \({\mathcal{F}}\) of Cartesian products of convex sets \(\prod_{i\in [t]}A_i\) in \({\mathbb{R}}^{td}\) with \(A_i\subset{\mathbb{R}}^d\), if at least \(\alpha \)-fraction of the \((d+1)\)-tuples in \({\mathcal{F}}\) are intersecting, then at least \((1-(t^d(1-\alpha ))^{1/(d+1)})\)-fraction of sets in \({\mathcal{F}}\) are intersecting. This is a special case of a more general result on intersections of \(d\)-Leray complexes. We also provide a construction showing that our result on \(d\)-Leray complexes is optimal. Interestingly, the extremal example is representable as a family of Cartesian products of convex sets, implying that the bound \(\alpha >1-1/t^d\) and the fraction \((1-(t^d(1-\alpha ))^{1/(d+1)})\) above are also best possible. The well-known optimal construction for fractional Helly theorem for convex sets in \({\mathbb{R}}^d\) does not have \((p,d+1)\)-condition for sublinear \(p\), that is, it contains a linear-size subfamily with no intersecting \((d+1)\)-tuple. Inspired by this, we give constructions showing that, somewhat surprisingly, imposing an additional \((p,d+1)\)-condition has a negligible effect on improving the quantitative bounds in neither the fractional Helly theorem for convex sets nor Cartesian products of convex sets. Our constructions offer a rich family of distinct extremal configurations for fractional Helly theorem, implying in a sense that the optimal bound is stable.On the volume ratio of projections of convex bodieshttps://zbmath.org/1529.520072024-04-02T17:33:48.828767Z"Galicer, Daniel"https://zbmath.org/authors/?q=ai:galicer.daniel"Litvak, Alexander E."https://zbmath.org/authors/?q=ai:litvak.alexander-e"Merzbacher, Mariano"https://zbmath.org/authors/?q=ai:merzbacher.mariano"Pinasco, Damián"https://zbmath.org/authors/?q=ai:pinasco.damianSummary: We study the volume ratio between \textit{projections} of two convex bodies. Given a high-dimensional convex body \(K\) we show that there is another convex body \(L\) such that the volume ratio between any two projections of fixed rank of the bodies \(K\) and \(L\) is large. Namely, we prove that for every \(1 \leq k \leq n\) and for each convex body \(K \subset \mathbb{R}^n\) there is a centrally symmetric body \(L \subset \mathbb{R}^n\) such that for any two projections \(P, Q : \mathbb{R}^n \to \mathbb{R}^n\) of rank \(k\) one has
\[
\begin{aligned}
&\operatorname{vr}(PK, QL) \geq c \min \left \{\frac{k}{\sqrt{n}} \sqrt{ \frac{ 1}{ \log \log \log (\frac{n \log (n)}{k})}} , \frac{ \sqrt{k}}{\sqrt{\log (\frac{ n \log (n)}{k} )}} \right\},
\end{aligned}
\]
where \(c > 0\) is an absolute constant. This general lower bound is sharp (up to logarithmic factors) in the regime \(k \geq n^{2 / 3}\).A negative answer to Ulam's problem 19 from the Scottish Bookhttps://zbmath.org/1529.520082024-04-02T17:33:48.828767Z"Ryabogin, Dmitry"https://zbmath.org/authors/?q=ai:ryabogin.dmitryIn this paper, the author answers in the negative an old problem posed by Ulam in the Scottish Book (as Problem 19), who asked if the three-dimensional (Euclidean) ball is a unique convex body \(K\) in the three-dimensional Euclidean space that floats in equilibrium in any orientation (in water of density \(1\)) provided it is made of material of uniform density strictly between \(0\) and \(1\). More generally, the author proves that in each dimension \(d \geq 3\) there exists a set \(K \subset \mathbb{R}^d\) that is a strictly convex non-centrally-symmetric body of revolution and floats in equilibrium in every orientation at the level \(\frac{|K|}{2}\) where \(|K|\) denotes the \(d\)-dimensional volume. As a consequence, he obtains a result that in \(\mathbb{R}^3\) there exists a convex body \(M\) of density \(\frac12\) that witnesses the failure of the Ulam's problem mentioned above. As was shown much earlier (by \textit{R. Schneider} [Enseign. Math. (2) 16, 297--305 (1971; Zbl 0209.26502)] and \textit{K. J. Falconer} [Am. Math. Mon. 90, 690--693 (1983; Zbl 0529.52001)]), no such \(M\) can be centrally symmetric. Other earlier results related with the topic of the paper are listed in the Introduction.
Reviewer: Piotr Niemiec (Kraków)Complete systems of inequalities relating the perimeter, the area and the Cheeger constant of planar domainshttps://zbmath.org/1529.520092024-04-02T17:33:48.828767Z"Ftouhi, Ilias"https://zbmath.org/authors/?q=ai:ftouhi.iliasThe article provides a complete characterization of the Blaschke-Santaló diagram for the triplet Cheeger constant, perimeter, and area of planar domains, describing inequalities involving such a triplet for several classes of sets: the simply connected sets, the convex sets, and the convex polygons with at most \(\mathrm{N}\) sides, where \(\mathrm{N}\) is at least 3. A quantitative version of the polygonal Faber-Krahn-type inequality for convex polygons is obtained.
Reviewer: Flavia-Corina Mitroi-Symeonidis (Bucureşti)Column-convex matrices, \(G\)-cyclic orders, and flow polytopeshttps://zbmath.org/1529.520102024-04-02T17:33:48.828767Z"González d'León, Rafael S."https://zbmath.org/authors/?q=ai:gonzalez-dleon.rafael-s"Hanusa, Christopher R. H."https://zbmath.org/authors/?q=ai:hanusa.christopher-r-h"Morales, Alejandro H."https://zbmath.org/authors/?q=ai:morales.alejandro-h"Yip, Martha"https://zbmath.org/authors/?q=ai:yip.marthaSummary: We study polytopes defined by inequalities of the form \(\sum_{i\in I}z_i\le 1\) for \(I\subseteq [d]\) and nonnegative \(z_i\) where the inequalities can be reordered into a matrix inequality involving a column-convex \(\{0,1\}\)-matrix. These generalize polytopes studied by Stanley, and the consecutive coordinate polytopes of Ayyer, Josuat-Vergès, and Ramassamy. We prove an integral equivalence between these polytopes and flow polytopes of directed acyclic graphs \(G\) with a Hamiltonian path, which we call spinal graphs. We show that the volumes of these flow polytopes are given by the number of upper (or lower) \(G\)-cyclic orders defined by the graphs \(G\). As a special case we recover results on volumes of consecutive coordinate polytopes. We study the combinatorics of \(k\)-Euler numbers, which are generalizations of the classical Euler numbers, and which arise as volumes of flow polytopes of a special family of spinal graphs. We show that their refinements, Ramassamy's \(k\)-Entringer numbers, can be realized as values of a Kostant partition function, satisfy a family of generalized boustrophedon recurrences, and are log concave along root directions. Finally, via our main integral equivalence and the known formula for the \(h^*\)-polynomial of consecutive coordinate polytopes, we give a combinatorial formula for the \(h^*\)-polynomial of flow polytopes of non-nested spinal graphs. For spinal graphs in general, we present a conjecture on upper and lower bounds for their \(h^*\)-polynomial.Throwing a sofa through the windowhttps://zbmath.org/1529.520112024-04-02T17:33:48.828767Z"Halperin, Dan"https://zbmath.org/authors/?q=ai:halperin.dan"Sharir, Micha"https://zbmath.org/authors/?q=ai:sharir.micha"Yehuda, Itay"https://zbmath.org/authors/?q=ai:yehuda.itaySummary: We study several variants of the problem of moving a convex polytope \(K\), with \(n\) edges, in three dimensions through a flat rectangular (and sometimes more general) window. Specifically: (i) We study variants where the motion is restricted to translations only, discuss situations where such a motion can be reduced to sliding (translation in a fixed direction), and present efficient algorithms for those variants, which run in time close to \(O(n^{8/3})\). (ii) We consider the case of a \textit{gate} (or a slab, an unbounded window with two parallel infinite edges), and show that \(K\) can pass through such a window, by any collision-free rigid motion, if and only if it can slide through it, an observation that leads to an efficient algorithm for this variant too. (iii) We consider arbitrary compact convex windows, and show that if \(K\) can pass through such a window \(W\) (by any motion) then \(K\) can slide through a gate of width equal to the diameter of \(W\). (iv) We show that if a purely translational motion for \(K\) through a rectangular window \(W\) exists, then \(K\) can also slide through \(W\) keeping the same orientation as in the translational motion. For a given fixed orientation of \(K\) we can determine in linear time whether \(K\) can translate (and hence slide) through \(W\) keeping the given orientation, and if so plan the motion, also in linear time. (v) We give an example of a polytope that cannot pass through a certain window by translations only, but can do so when rotations are allowed. (vi) We study the case of a circular window \(W\), and show that, for the regular tetrahedron \(K\) of edge length 1, there are two thresholds \(1> \delta_1\approx 0.901388 > \delta_2\approx 0.895611\), such that (a) \(K\) can slide through \(W\) if the diameter \(d\) of \(W\) is \(\geqslant 1\), (b) \(K\) cannot slide through \(W\) but can pass through it by a purely translational motion when \(\delta_1 \leq d < 1\), (c) \(K\) cannot pass through \(W\) by a purely translational motion but can do it when rotations are allowed when \(\delta_2 \leq d < \delta_1\), and (d) \(K\) cannot pass through \(W\) at all when \(d < \delta_2\). (vii) Finally, we explore the general setup, where we want to plan a general motion (with all six degrees of freedom) for \(K\) through a rectangular window \(W\), and present an efficient algorithm for this problem, with running time close to \(O(n^4)\).\(P\)-associahedrahttps://zbmath.org/1529.520122024-04-02T17:33:48.828767Z"Galashin, Pavel"https://zbmath.org/authors/?q=ai:galashin.pavelSummary: For each poset \(P\), we construct a polytope \(\mathscr{A}(P)\) called the \(P\)-\textit{associahedron}. Similarly to the case of graph associahedra, the faces of \(\mathscr{A}(P)\) correspond to certain nested collections of subsets of \(P\). The Stasheff associahedron is a compactification of the configuration space of \(n\) points on a line, and we recover \(\mathscr{A}(P)\) as an analogous compactification of the space of order-preserving maps \(P\rightarrow\mathbb{R}\). Motivated by the study of totally nonnegative critical varieties in the Grassmannian, we introduce \textit{affine poset cyclohedra} and realize these polytopes as compactifications of configuration spaces of \(n\) points on a circle. For particular choices of (affine) posets, we obtain associahedra, cyclohedra, permutohedra, and type \(B\) permutohedra as special cases.Gram's equation -- a probabilistic proofhttps://zbmath.org/1529.520132024-04-02T17:33:48.828767Z"Welzl, Emo"https://zbmath.org/authors/?q=ai:welzl.emoFor the entire collection see [Zbl 0875.00071].Pizza and 2-structureshttps://zbmath.org/1529.520142024-04-02T17:33:48.828767Z"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: Let \({\mathcal{H}}\) be a Coxeter hyperplane arrangement in \(n\)-dimensional Euclidean space. Assume that the negative of the identity map belongs to the associated Coxeter group \(W\). Furthermore assume that the arrangement is not of type \(A_1^n\). Let \(K\) be a measurable subset of the Euclidean space with finite volume which is stable by the Coxeter group \(W\) and let \(a\) be a point such that \(K\) contains the convex hull of the orbit of the point \(a\) under the group \(W\). In a previous article the authors proved the generalized pizza theorem: that the alternating sum over the chambers \(T\) of \({\mathcal{H}}\) of the volumes of the intersections \(T\cap (K+a)\) is zero. In this paper we give a dissection proof of this result. In fact, we lift the identity to an abstract dissection group to obtain a similar identity that replaces the volume by any valuation that is invariant under affine isometries. This includes the cases of all intrinsic volumes. Apart from basic geometry, the main ingredient is a theorem of the authors where we relate the alternating sum of the values of certain valuations over the chambers of a Coxeter arrangement to similar alternating sums for simpler subarrangements called 2-structures introduced by Herb to study discrete series characters of real reduced groups.Monotone paths on cross-polytopeshttps://zbmath.org/1529.520152024-04-02T17:33:48.828767Z"Black, Alexander E."https://zbmath.org/authors/?q=ai:black.alexander-e"De Loera, Jesús A."https://zbmath.org/authors/?q=ai:de-loera.jesus-aSummary: In the early 1990s, Billera and Sturmfels introduced the monotone path polytope (MPP), an important case of the general theory of fiber polytopes, which has led to remarkable combinatorics. Given a pair \((P,\varphi )\) of a polytope \(P\) and a linear functional \(\varphi \), the MPP is obtained from averaging the fibers of the projection \(\varphi (P)\). They also showed that MPPs of (regular) simplices and hyper-cubes are combinatorial cubes and permutahedra respectively. As a natural follow-up we investigate the monotone paths of cross-polytopes for a generic linear functional \(\varphi \). We show the face lattice of the MPP of the cross-polytope is isomorphic to the lattice of intervals in the sign poset from oriented matroid theory. We also describe its \(f\)-vector, some geometric realizations, an irredundant inequality description, the 1-skeleton and we compute its diameter. In contrast to the case of simplices and hyper-cubes, monotone paths on cross-polytopes are not always coherent.Distinct angle problems and variantshttps://zbmath.org/1529.520162024-04-02T17:33:48.828767Z"Fleischmann, Henry L."https://zbmath.org/authors/?q=ai:fleischmann.henry-l"Hu, Hongyi B."https://zbmath.org/authors/?q=ai:hu.hongyi-b"Jackson, Faye"https://zbmath.org/authors/?q=ai:jackson.faye"Miller, Steven J."https://zbmath.org/authors/?q=ai:miller.steven-j"Palsson, Eyvindur A."https://zbmath.org/authors/?q=ai:palsson.eyvindur-ari"Pesikoff, Ethan"https://zbmath.org/authors/?q=ai:pesikoff.ethan"Wolf, Charles"https://zbmath.org/authors/?q=ai:wolf.charlesSummary: The Erdős distinct distance problem is a ubiquitous problem in discrete geometry. Less well known is Erdős's distinct angle problem, the problem of finding the minimum number of distinct angles between \(n\) non-collinear points in the plane. The standard problem is already well understood. However, it admits many of the same variants as the distinct distance problem, many of which are unstudied. We provide upper and lower bounds on a broad class of distinct angle problems. We show that the number of distinct angles formed by \(n\) points in general position is \(O(n^{\log_27})\) providing the first non-trivial bound for this quantity. We introduce a new class of asymptotically optimal point configurations with no four cocircular points. Then, we analyze the sensitivity of asymptotically optimal point sets to perturbation, yielding a much broader class of asymptotically optimal configurations. In higher dimensions we show that a variant of Lenz's construction admits fewer distinct angles than the optimal configurations in two dimensions. We also show that the minimum size of a maximal subset of \(n\) points in general position admitting only unique angles is \(\Omega (n^{1/5})\) and \(O(n^{(\log_2{7)/3}})\). We also provide bounds on the partite variants of the standard distinct angle problem.Optimal point sets determining few distinct angleshttps://zbmath.org/1529.520172024-04-02T17:33:48.828767Z"Fleischmann, Henry L."https://zbmath.org/authors/?q=ai:fleischmann.henry-l"Miller, Steven J."https://zbmath.org/authors/?q=ai:miller.steven-j"Palsson, Eyvindur A."https://zbmath.org/authors/?q=ai:palsson.eyvindur-ari"Pesikoff, Ethan"https://zbmath.org/authors/?q=ai:pesikoff.ethan"Wolf, Charles"https://zbmath.org/authors/?q=ai:wolf.charlesThe paper under review investigates the following problem: what is the maximum size \(P(k)\) of a finite point set in the plane, such that not all points are on a line, and the point set produces exactly \(k\) distinct angles that lie in \((0,\pi)\)? The paper shows that for all \(k\geq 1\), \(2k + 3\leq P(2k)\leq 12k\) and \(2k + 3\leq P(2k + 1)\leq 12k+6\), and makes the conjecture that \(P(k)\) equals to the lower bound.
Reviewer: László A. Székely (Columbia)On the cardinality of sets in \(\mathbb{R}^d\) obeying a slightly obtuse angle boundhttps://zbmath.org/1529.520182024-04-02T17:33:48.828767Z"Lim, Tongseok"https://zbmath.org/authors/?q=ai:lim.tongseok"McCann, Robert J."https://zbmath.org/authors/?q=ai:mccann.robert-jPaul Erdős posed the problem that turned into the following classical theorem: if in a finite point set \(A\subset \mathbb{R}^d\) no triangle with vertices from \(A\) has an angle exceeding \(\pi/2\), then \(|A|\leq 2^d\). The paper under review investigates finite points set \(A\subset \mathbb{R}^d\), in which every angle is strictly less than \(\arccos(-1/d)\in (\pi/2,\pi)\). It is shown that \(|A\) is finite, an explicit bound is found, furthermore, the points of \(A\) make the vertices of a convex polytope.
Reviewer: László A. Székely (Columbia)Undecidable translational tilings with only two tiles, or one nonabelian tilehttps://zbmath.org/1529.520192024-04-02T17:33:48.828767Z"Greenfeld, Rachel"https://zbmath.org/authors/?q=ai:greenfeld.rachel"Tao, Terence"https://zbmath.org/authors/?q=ai:tao.terence-cSummary: We construct an example of a group \(G = \mathbb{Z}^2 \times G_0\) for a finite abelian group \(G_0\), a subset \(E\) of \(G_0\), and two finite subsets \(F_1,F_2\) of \(G\), such that it is undecidable in ZFC whether \(\mathbb{Z}^2\times E\) can be tiled by translations of \(F_1,F_2\). In particular, this implies that this tiling problem is \textit{aperiodic}, in the sense that (in the standard universe of ZFC) there exist translational tilings of \(E\) by the tiles \(F_1,F_2\), but no periodic tilings. Previously, such aperiodic or undecidable translational tilings were only constructed for sets of eleven or more tiles (mostly in \(\mathbb{Z}^2\)). A similar construction also applies for \(G=\mathbb{Z}^d\) for sufficiently large \(d\). If one allows the group \(G_0\) to be non-abelian, a variant of the construction produces an undecidable translational tiling with only one tile \(F\). The argument proceeds by first observing that a single tiling equation is able to encode an arbitrary system of tiling equations, which in turn can encode an arbitrary system of certain functional equations once one has two or more tiles. In particular, one can use two tiles to encode tiling problems for an arbitrary number of tiles.Corrigendum to: ``Flexible circuits in the \(d\)-dimensional rigidity matroid''https://zbmath.org/1529.520202024-04-02T17:33:48.828767Z"Grasegger, Georg"https://zbmath.org/authors/?q=ai:grasegger.georg"Guler, Hakan"https://zbmath.org/authors/?q=ai:guler.hakan"Jackson, Bill"https://zbmath.org/authors/?q=ai:jackson.bill"Nixon, Anthony"https://zbmath.org/authors/?q=ai:nixon.anthonyFrom the text: We give a counter example to Lemma 18(a) and Conjecture 17 in our paper [\textit{G. Grasegger} et al., ibid. 100, No. 2, 315--330 (2022; Zbl 1523.52032)] and provide a corrected proof for a weaker version of Lemma 18(a).Equilibrium stresses and rigidity for infinite tensegrities and frameworkshttps://zbmath.org/1529.520212024-04-02T17:33:48.828767Z"Power, S. C."https://zbmath.org/authors/?q=ai:power.stephen-cA finite bar-joint framework in Euclidean \(d\)-space is a structure consisting of rigid bars, which are connected at their endpoints in the manner of a linearly embedded graph. A finite tensegrity is a structure consisting of inextensible cables, incompressible struts and rigid bars, which are connected at their endpoints. The author studies countably infinite tensegrities. \par Amid the main results of the article we mention characterisations of the first-order rigidity of countably infinite tensegrities relative to asymptotically decaying motions, such as first-order \(c_0\)-rigidity, for velocity fields that tend to zero at infinity, and first-order \({\ell}^2\)-rigidity, for finite energy velocity fields. These results generalize the characterisation of the first-order rigidity of a finite tensegrity obtained by \textit{B. Roth} and \textit{W. Whiteley} in [Trans. Am. Math. Soc. 265, 419--446 (1981; Zbl 0479.51015)].
The author also considers equilibrium stresses from the point of view of generalising the notion of prestress stability to countably infinite tensegrities and bar-joint frameworks.
The proofs are based on a new short proof for finite bar-joint frameworks that prestress stability ensures continuous rigidity.
Reviewer: Victor Alexandrov (Novosibirsk)An identity for the coefficients of characteristic polynomials of hyperplane arrangementshttps://zbmath.org/1529.520222024-04-02T17:33:48.828767Z"Kabluchko, Zakhar"https://zbmath.org/authors/?q=ai:kabluchko.zakhar-aSummary: Consider a finite collection of affine hyperplanes in \(\mathbb{R}^d\). The hyperplanes dissect \(\mathbb{R}^d\) into finitely many polyhedral chambers. For a point \(x\in \mathbb{R}^d\) and a chamber \(P\) the metric projection of \(x\) onto \(P\) is the unique point \(y\in P\) minimizing the Euclidean distance to \(x\). The metric projection is contained in the relative interior of a uniquely defined face of \(P\) whose dimension is denoted by \(\dim(x,P)\). We prove that for every given \(k\in \{0,\ldots , d\} \), the number of chambers \(P\) for which \(\dim(x,P) = k\) does not depend on the choice of \(x\), with an exception of some Lebesgue null set. Moreover, this number is equal to the absolute value of the \(k\)-th coefficient of the characteristic polynomial of the hyperplane arrangement. In a special case of reflection arrangements, this proves a conjecture of \textit{M. Drton} and \textit{C. J. Klivans} [Proc. Am. Math. Soc. 138, No. 8, 2873--2887 (2010; Zbl 1208.51008)].New weighted geometric inequalities for hypersurfaces in space formshttps://zbmath.org/1529.530862024-04-02T17:33:48.828767Z"Wei, Yong"https://zbmath.org/authors/?q=ai:wei.yong"Zhou, Tailong"https://zbmath.org/authors/?q=ai:zhou.tailongThe authors prove new sharp geometric inequalities involving weighted curvature integrals and quermassintegrals for smooth closed hypersurfaces in space forms. After reviewing some properties of elementary symmetric functions and variational equations, they apply the inverse curvature flow to show one of the main results, namely, to prove a complete family of sharp weighted geometric inequalities for a smooth, closed, star-shaped and \(k\)-convex hypersurface in an Euclidean space enclosing a bounded domain, in a hyperbolic space and in a sphere. The tools they use are the inverse curvature flow by \textit{C. Gerhardt} [J. Differ. Geom. 32, No. 1, 299--314 (1990; Zbl 0708.53045)] and \textit{J. I. E. Urbas} [Math. Z. 205, No. 3, 355--372 (1990; Zbl 0691.35048)] and the locally constrained curvature flows introduced by \textit{S. Brendle, P. Guan} and \textit{J. Li} [Preprint, 2018].
Reviewer: Adara M. Blaga (Timişoara)Long lines in subsets of large measure in high dimensionhttps://zbmath.org/1529.600152024-04-02T17:33:48.828767Z"Elboim, Dor"https://zbmath.org/authors/?q=ai:elboim.dor"Klartag, Bo'az"https://zbmath.org/authors/?q=ai:klartag.boazA very interesting finding of the probability theory is established: for any set \(A\subseteq [0, 1]^n\) with \(\mathrm{Vol}(A)\geq 1/2\) there exists a line \(l\) such that the one-dimensional Lebesgue measure of \(l\cap A\) is at least \(\Omega(n^{1/4})\). The exponent 1/4 is tight. For a probability measure \(\mu\) on \(\mathbb R^n\) and \(0<a<1\) define
\[
L(\mu, a):=\inf_{A;\mu(A)=a}\sup_{l \text{ line}} l\cap A.
\]
where \(|\cdot|\) stands for the one-dimensional Lebesgue measure. The asymptotic behavior of \(L(\mu, a)\) is studied.
Very nice figures, well-developed theorems: this paper is an excellent application of the theory of probability.
Reviewer: Rózsa Horváth-Bokor (Budakalász)Phase transition for the volume of high-dimensional random polytopeshttps://zbmath.org/1529.600202024-04-02T17:33:48.828767Z"Bonnet, Gilles"https://zbmath.org/authors/?q=ai:bonnet.gilles"Kabluchko, Zakhar"https://zbmath.org/authors/?q=ai:kabluchko.zakhar-a"Turchi, Nicola"https://zbmath.org/authors/?q=ai:turchi.nicolaThe beta polytope \(P_{n,d}^\beta\) is the convex hull of \(n\) i.i.d.\ points distributed in the unit Euclidean ball in \(\mathbb{R}^d\) according to the density proportional to \((1-\|x\|^2)^\beta\) with \(\beta>-1\) and uniform on the boundary of the ball if \(\beta=-1\).
The authors study the behaviour of beta polytopes in high dimensions. They show that the expected \(d\)-volume of the beta polytope normalised by the volume of the unit ball exhibits a phase transition depending on the growth regime of the number of points in relation to the dimension. Analogous results are obtained for the intrinsic volumes and, when \(\beta=0\), for the number of vertices.
Reviewer: Ilya S. Molchanov (Bern)Limit theory for the first layers of the random convex hull peeling in the unit ballhttps://zbmath.org/1529.600212024-04-02T17:33:48.828767Z"Calka, Pierre"https://zbmath.org/authors/?q=ai:calka.pierre"Quilan, Gauthier"https://zbmath.org/authors/?q=ai:quilan.gauthierA very interesting study is performed on the law of large numbers for the first layers of the random convex hull peeling in the unit ball. The figures and items are very nicely done and easy to read.
Reviewer: Rózsa Horváth-Bokor (Budakalász)Towards lower bounds on the depth of ReLU neural networkshttps://zbmath.org/1529.682762024-04-02T17:33:48.828767Z"Hertrich, Christoph"https://zbmath.org/authors/?q=ai:hertrich.christoph"Basu, Amitabh"https://zbmath.org/authors/?q=ai:basu.amitabh"Di Summa, Marco"https://zbmath.org/authors/?q=ai:di-summa.marco"Skutella, Martin"https://zbmath.org/authors/?q=ai:skutella.martinSummary: We contribute to a better understanding of the class of functions that can be represented by a neural network with ReLU activations and a given architecture. Using techniques from mixed-integer optimization, polyhedral theory, and tropical geometry, we provide a mathematical counterbalance to the universal approximation theorems which suggest that a single hidden layer is sufficient for learning any function. In particular, we investigate whether the class of exactly representable functions strictly increases by adding more layers (with no restrictions on size). As a by-product of our investigations, we settle an old conjecture about piecewise linear functions by \textit{S. Wang} and \textit{X. Sun} [IEEE Trans. Inf. Theory 51, No. 12, 4425--4431 (2005; Zbl 1283.94021)] in the affirmative. We also present upper bounds on the sizes of neural networks required to represent functions with logarithmic depth.Extremal transitions via quantum Serre dualityhttps://zbmath.org/1529.810472024-04-02T17:33:48.828767Z"Mi, Rongxiao"https://zbmath.org/authors/?q=ai:mi.rongxiao"Shoemaker, Mark"https://zbmath.org/authors/?q=ai:shoemaker.markSummary: Two varieties \(Z\) and \({\widetilde{Z}}\) are said to be related by extremal transition if there exists a degeneration from \(Z\) to a singular variety \({\overline{Z}}\) and a crepant resolution \({\widetilde{Z}} \rightarrow{\overline{Z}}\). In this paper we compare the genus-zero Gromov-Witten theory of toric hypersurfaces related by extremal transitions arising from toric blow-up. We show that the quantum \(D\)-module of \({\widetilde{Z}}\), after analytic continuation and restriction of a parameter, recovers the quantum \(D\)-module of \(Z\). The proof provides a geometric explanation for both the analytic continuation and restriction parameter appearing in the theorem.Black holes in a new gravitational theory with trace anomalieshttps://zbmath.org/1529.830662024-04-02T17:33:48.828767Z"Tsujikawa, Shinji"https://zbmath.org/authors/?q=ai:tsujikawa.shinjiSummary: In a new gravitational theory with the trace anomaly recently proposed by Gabadadze, we study the existence of hairy black hole solutions on a static and spherically symmetric background. In this theory, the effective 4-dimensional action contains a kinetic term of the conformal scalar field related to a new scale \(\overline{M}\) much below the Planck mass. This property can overcome a strong coupling problem known to be present in general relativity supplemented by the trace anomaly as well as in 4-dimensional Einstein-Gauss-Bonnet gravity. We find a new hairy black hole solution arising from the Gauss-Bonnet trace anomaly, which satisfies regular boundary conditions of the conformal scalar and metric on the horizon. Unlike unstable exact black hole solutions with a divergent derivative of the scalar on the horizon derived for some related theories in the literature, we show that our hairy black hole solution can be consistent with all the linear stability conditions of odd- and even-parity perturbations.Testing new massive conformal gravity with the light deflection by black holehttps://zbmath.org/1529.830682024-04-02T17:33:48.828767Z"Yasir, Muhammad"https://zbmath.org/authors/?q=ai:yasir.muhammad"Tiecheng, Xia"https://zbmath.org/authors/?q=ai:tiecheng.xia"Mushtaq, Farzan"https://zbmath.org/authors/?q=ai:mushtaq.farzan"Bamba, Kazuharu"https://zbmath.org/authors/?q=ai:bamba.kazuharuSummary: We study the weak gravitational lensing effect of Non-Bocharova-Bronnikov-Melnikov-Bekenstein (BBMB) black hole in new massive conformal gravity. We analyze the deflection angle of light caused by new massive conformal gravity by using the Gauss-Bonnet theorem. As a consequence, we obtain the Gaussian optical curvature and calculate the deflection angle of the new massive conformal gravity for spherically balanced space-time with the Gauss-Bonnet theorem. The resultant deflection angle of light in the weak field limits showing that the bending of light is a global and topological phenomenon. Furthermore, we identify the deflection angle of light in the framework of the plasma medium, we also demonstrate the effect of a plasma medium on the deflection of light by non BBMB. In addition, the behavior of the deflection angle by new massive conformal gravity is explicitly shown in the influence of plasma medium and for the non plasma medium.Enumeration and unimodular equivalence of empty delta-modular simpliceshttps://zbmath.org/1529.900522024-04-02T17:33:48.828767Z"Gribanov, D. V."https://zbmath.org/authors/?q=ai:gribanov.dmitrii-vladimirovich|gribanov.dimitry-vSummary: Consider a class of simplices defined by systems \(A x \le b\) of linear inequalities with \(\varDelta \)-\textit{modular} matrices. A matrix is called \(\varDelta \)-\textit{modular}, if all its rank-order sub-determinants are bounded by \(\varDelta\) in an absolute value. In our work we call a simplex \(\varDelta \)-\textit{modular}, if it can be defined by a system \(A x \le b\) with a \(\varDelta \)-\textit{modular} matrix A. And we call a simplex \textit{empty}, if it contains no points with integer coordinates. In literature, a simplex is called \textit{lattice}, if all its vertices have integer coordinates. And a lattice-simplex is called \textit{empty}, if it contains no points with integer coordinates excluding its vertices. Recently, assuming that \(\varDelta\) is fixed, it was shown in [\textit{D. Gribanov} et al., ``On \(\Delta\)-modular integer linear problems in the canonical form and equivalent problems'', J. Glob. Optim. 2022, 61 p. (2022; \url{doi:10.1007/s10898-022-01165-9})] that the number of \(\varDelta \)-\textit{modular} empty simplices modulo the unimodular equivalence relation is bounded by a polynomial on dimension. We show that an analogous fact holds for the class of \(\varDelta \)-\textit{modular} empty lattice-simplices. As the main result, assuming again that the value of the parameter \(\varDelta\) is fixed, we show that the all unimodular equivalence classes of simplices of both types can be enumerated by a polynomial-time algorithm. As the secondary result, we show the existence of a polynomial-time algorithm for the problem to check the unimodular equivalence relation for a given pair of \(\varDelta \)-\textit{modular} (not necessarily empty) simplices.
For the entire collection see [Zbl 1517.90002].