Recent zbMATH articles in MSC 52https://zbmath.org/atom/cc/522021-01-08T12:24:00+00:00WerkzeugEhrhart polynomial for lattice squares, cubes and hypercubes.https://zbmath.org/1449.520102021-01-08T12:24:00+00:00"Ionascu, Eugen J."https://zbmath.org/authors/?q=ai:ionascu.eugen-julienFor a given \(d\)-dimensional compact simplicial complex \(\mathcal{P}\) in \({\mathbb{R}}^n\), which has all its vertices in the integer lattice, the number of lattice points in \(t\mathcal{P}\) for a positive integer \(t\) is given by a rational polynomial \(L({\mathcal{P}},t)\) of degree \(d\) of the variable \(t\). This well studied polynomial is called the ``Ehrhart polynomial'' of the simplicial complex. It is known that the number of lattice points in the interior of \(t\mathcal{P}\) is expressed by the Ehrhart polynomial as \((-1)^d L({\mathcal{P}},-t)\). For \(n=2\), the paper under review studies the sequence of integers that arise as number of latttice points in the interior of a lattice square, put into increasing order. This sequence is called ``almost perfect squares''. The paper observes that this sequence is very similar, but not identical, to the sequence A194154 of the On-Line Encyclopedia of Integer Sequences. The paper constructs integer lattice squares, cubes, and hypercubes in \(n=2,3,4\) dimensions. All integer lattice squares are characterized in \({\mathbb{R}}^4\). For each of these problems, the corresponding Ehrhart polynomial is computed.
Reviewer: László A. Székely (Columbia)An analogue of Sylvester's four-point problem on the sphere.https://zbmath.org/1449.600132021-01-08T12:24:00+00:00"Maehara, H."https://zbmath.org/authors/?q=ai:maehara.hiroshi"Martini, H."https://zbmath.org/authors/?q=ai:martini.horstSummary: A finite subset \(X\) of the unit sphere \(\mathbb{S}^{d-1}\) in \(\mathbb{R}^d\) is called extremal if, for every \(x\in X\), there is a hemisphere that contains \(X\setminus \{x\}\) in its interior and has \(x\) on its boundary. Let \(P\) denote the probability that a random sample of \(d+1\) points, chosen uniformly from \(\mathbb{S}^{d-1}\), is extremal. We show that \(P=1-(d+2)/2^d\).On the reverse Orlicz-Lorentz Busemann-Petty centroid inequality.https://zbmath.org/1449.520042021-01-08T12:24:00+00:00"Feng, Y."https://zbmath.org/authors/?q=ai:feng.yuquan|feng.yuanrui|feng.yani|feng.yejuan|feng.yarong|feng.yuming|feng.yujuan|feng.yingjun|feng.yongbao|feng.yulin|feng.yuying|feng.yunfei|feng.yougang|feng.yue|feng.yilan|feng.yuyou|feng.yanquan|feng.yuyu|feng.yunqing|feng.yinbo|feng.yuehua|feng.yanping|feng.yuhu|feng.youyi|feng.yaqin|feng.yongxun|feng.yuanming|feng.yin|feng.yongjie|feng.yaodong|feng.yijia|feng.yuling|feng.yuxu|feng.yan|feng.yingjie|feng.yinchuan|feng.yuang|feng.yingtao|feng.yinshan|feng.yunwen|feng.yifu|feng.yifei|feng.yuxing|feng.yanfei|feng.yongqi|feng.yuee|feng.yanzhao|feng.yuchun|feng.yuanli|feng.yiying|feng.yuanping|feng.yijun|feng.yuanfu|feng.yuanjian|feng.yidong|feng.yongjin|feng.yuankun|feng.yongxin|feng.yangde|feng.yibin|feng.yuncheng|feng.yiwei|feng.yonghui|feng.yangang|feng.yanhong|feng.yongliang|feng.yunyun|feng.yun|feng.youji|feng.yiping|feng.yushu|feng.yanji|feng.yuan|feng.yusheng|feng.yali|feng.yanghe|feng.yongping|feng.yu|feng.yingying|feng.youwen|feng.yanbin|feng.yaping|feng.yanyan|feng.yuecai|feng.yanqin|feng.yongde|feng.yonghan|feng.yuehong|feng.yueping|feng.yuqiang|feng.yanfa|feng.yucai|feng.yanxiang|feng.yankun|feng.yucheng|feng.yuanzhen|feng.yonge|feng.yanqing|feng.ying|feng.yuansheng|feng.yizhuo|feng.ye|feng.yumei|feng.yulu|feng.yingling|feng.yiteng|feng.youran|feng.yuntian|feng.yuzhang|feng.yanying|feng.yi|feng.yanling|feng.yinjiang|feng.yiyong|feng.yaoyao|feng.yang|feng.yong|feng.yunzhi|feng.yumin|feng.yixiong|feng.yuefeng|feng.yongqing|feng.yuanyuan|feng.yongchang|feng.yuhong|feng.yuanjing|feng.yiliu|feng.yuanhua|feng.yaning|feng.youyong|feng.yao|feng.yihu|feng.yuqing|feng.youqian|feng.yanming|feng.yantao|feng.youhe|feng.yinqi|feng.yanan|feng.youling|feng.yujing|feng.yunlong|feng.yuting|feng.yansong"Ma, T."https://zbmath.org/authors/?q=ai:ma.tianyu|ma.timmy|ma.teng|ma.thomas|ma.tongshu|ma.tianyun|ma.to-fu|ma.tianshui|ma.tiedong|ma.tiantian|ma.tan|ma.tingfu|ma.tengcai|ma.ting|ma.tianyi|ma.tieyou|ma.tianlong|ma.tianzhou|ma.tianxing|ma.tieju|ma.tongyi|ma.tianyang|ma.tinghuai|ma.tian|ma.tianlei|ma.tong|ma.tsoywo|ma.tianhao|ma.tao|ma.tiejun|ma.tiefeng|ma.tianshan|ma.tianbao|ma.tengteng|ma.tingfeng|ma.tengyu|ma.tongze|ma.tianxiao|ma.tongy|ma.tingting|ma.tianfuSummary: This paper studies the extrema of some affine invariant functionals related to the volume of the Orlicz-Lorentz centroid body introduced by \textit{V. H. Nguyen} [Adv. Appl. Math. 92, 99--121 (2018; Zbl 1380.52011)]. We obtain some variants of the Orlicz-Lorentz Busemann-Petty centroid inequality, and also prove the reverse form of these inequalities in the two-dimensional case.Characteristic polynomials of generic arrangements.https://zbmath.org/1449.050402021-01-08T12:24:00+00:00"Sun, Ying"https://zbmath.org/authors/?q=ai:sun.ying"Gao, Ruimei"https://zbmath.org/authors/?q=ai:gao.ruimeiSummary: Firstly, we gave the definition of the generalized general position arrangement, and studied the relationships among the general position arrangements, generalized general position arrangements and generic arrangements. Secondly, by establishing the relationships between arrangements and the simple graphs, we gave the necessary and sufficient condition for linear independence of the subarrangements of generic threshold arrangements and the characteristic polynomials of the subarrangements of generic threshold arrangements.Tangle-tree duality: in graphs, matroids and beyond.https://zbmath.org/1449.051562021-01-08T12:24:00+00:00"Diestel, Reinhard"https://zbmath.org/authors/?q=ai:diestel.reinhard"Oum, Sang-Il"https://zbmath.org/authors/?q=ai:oum.sang-ilThe authors build upon a general width duality theorem in abstract separation systems with well-defined notions of cohesion and separation, which establishes duality between the existence of high cohesiveness at a local level and a global tree structure, and which they have proved recently by the authors [``Tangle-tree duality in abstract separation systems'', Preprint, \url{arXiv:1701.02509}]. In the present paper this theorem is applied to derive tangle-tree-type duality theorems for tree-width, path-width, tree-decompositions of small adhesion in graphs, for tree-width in matroids, and its application to data science is showcased through derivation of the duality theorem for the existence of clusters in large data sets. Its power is further demonstrated by obtaining the classical duality theorems for width parameters in graph minor theory (such as path-width, tree-width, branch-width and rank-width) as its corollaries.
Reviewer: Dragan Stevanović (Niš)Tiling convex polygons with congruent isosceles right triangles.https://zbmath.org/1449.520122021-01-08T12:24:00+00:00"Chen, Hongjing"https://zbmath.org/authors/?q=ai:chen.hongjing"Su, Zhanjun"https://zbmath.org/authors/?q=ai:su.zhanjun"Zhang, Xiaopeng"https://zbmath.org/authors/?q=ai:zhang.xiaopengSummary: Let \({P_v}\) be a plane convex polygon with \(v\) vertices. It was proved that \({P_v}\) can be tiled with congruent equilateral triangles for \(v = 3, 4, 5, 6\). In this paper we consider the corresponding problem for isosceles right triangles and prove that if \({P_v}\) can be cut into finitely many congruent isosceles right triangles then \(v = 3, 4, 5, 6, 7, 8\). In particular, we determine the sets \(T = \{ (v,k):v \in \{3, 4, 5, \cdots \}, k \in \{1, 2, 3, \cdots\}\}\), and there exists a convex \(v\)-gon that can be tiled with \(k\) congruent isosceles right triangles. In particular, our results on tilings of pentagons are essentially based on a relation with the so-called idoneal numbers.A new upper bound for the size of \(s\)-distance sets in boxes.https://zbmath.org/1449.520112021-01-08T12:24:00+00:00"Hegedűs, G."https://zbmath.org/authors/?q=ai:hegedus.gaborThe subject is related to the following distance problem by Erdős: Given \(n\) points in the plane, what is the smallest number of distinct distances they can determine? In 1946 Erdős gave some inequalities for the minimum number of these distances.
An \(s\)-distance set is any subset \(G\) of \(n\)-dimensional Euclidean space such that the size of the set of distances among points of \(G\) (the number of distinct distances) is at most \(s\).
In the paper, the author investigates \(s\)-distance sets in the direct product of finite sets of points (boxes) in \(n\)-dimensional Euclidean space. A new upper bound for the size of \(s\)-distance sets in boxes is given. In the proof, the author uses Tao's slice rank method.
Reviewer: Elizaveta Zamorzaeva (Chişinău)Fixed point theorems on locally T-convex spaces.https://zbmath.org/1449.540572021-01-08T12:24:00+00:00"Chen, Zhiyou"https://zbmath.org/authors/?q=ai:chen.zhiyouSummary: In this paper, in locally T-convex spaces, two new fixed point theorems are established without using the KKM technique, which respectively take the famous Schauder's and Browder's fixed point theorems on H-spaces as their particular cases, so that these two important theorems are generalized to T-convex spaces.Circles and crossing planar compact convex sets.https://zbmath.org/1449.520022021-01-08T12:24:00+00:00"Czédli, Gábor"https://zbmath.org/authors/?q=ai:czedli.gaborSummary: Let \(K_0\) be a compact convex subset of the plane \(\mathbb{R}^2\), and assume that whenever \(K_1\subseteq \mathbb{R}^2\) is congruent to \(K_0\), then \(K_0\) and \(K_1\) are not crossing in a natural sense due to L. Fejes-Tóth. A theorem of \textit{L. Fejes-Tóth} [Elem. Math. 22, 25--27 (1967; Zbl 0153.52002)] states that the assumption above holds for \(K_0\) if and only if \(K_0\) is a disk. \textit{G. Czédli} [Acta Sci. Math. 83, No. 3--4, 703--712 (2017; Zbl 1413.52001)] introduced a new concept of crossing, and proved that L. Fejes-Tóth's theorem remains true if the old concept is replaced by the new one. Our purpose is to describe the hierarchy among several variants of the new concepts and the old concept of crossing. In particular, we prove that each variant of the new concept of crossing is more restrictive than the old one. Therefore, L. Fejes-Tóth's theorem from 1967 becomes an immediate consequence of the 2017 characterization of circles but not conversely. Finally, a mini-survey shows that this purely geometric paper has precursors in combinatorics and, mainly, in lattice theory.Area minimization of special polygons.https://zbmath.org/1449.520032021-01-08T12:24:00+00:00"Bezdek, A."https://zbmath.org/authors/?q=ai:bezdek.andras"Joós, A."https://zbmath.org/authors/?q=ai:joos.antalThe paper under review concerns the following statement on a particular area minimization problem for convex polygons.
Theorem (Hajós lemma). Fix a circle \(\gamma\) inside a concentric unit circle \(\Gamma\). Among all convex polygons which contain the smaller circle \(\gamma\) and have no vertices inside the unit circle \(\Gamma\), the polygon which is inscribed in \(\Gamma\) so that all its sides, with the exception of at most one side, is tangent to \(\gamma\) is of minimum area.
The authors extend Hajós lemma to the case of non-concentric circles as follows.
Theorem. Fix a small circle \(\gamma\) inside a unit circle \(\Gamma\) and assume that \(\gamma\) is non-concentric to \(\Gamma\). Consider all polygons which are inscribed in the unit circle \(\Gamma\) and contain the smaller circle \(\gamma\). Then the smallest area polygon has all its sides, with the exception of at most one side, tangent to \(\gamma\).
Moreover, Hajós lemma is extended to the case of specific planar convex sets called disc polygons. By definition, an \(R\)-disc polygon is the intersection of finitely many congruent discs of radius \(R\). The boundary of a disc polygon consists of circular arcs called sides, the common points of adjacent sides are called vertices. An \(R\)-disc polygon with \(R>1\) is said to be inscribed in a unit circle if all its vertices lie on this unit circle. Then the following version of Hajós lemma is proved.
Theorem. Let \(0<r<1\) and \(R>1\). Fix a circle \(\gamma\) of radius \(r\) inside of a concentric unit circle \(\Gamma\). Among all \(R\)-disc polygons which contain the circle \(\gamma\) and have no vertices inside the unit circle \(\Gamma\), the \(R\)-disc polygon which is inscribed in \(\Gamma\) so that all its sides, with the exception of at most one side, are tangent to \(\gamma\) is of minimum area.
Reviewer: Vasyl Gorkaviy (Kharkov)New Brunn-Minkowski type inequalities for general width-integral of index \(i\).https://zbmath.org/1449.260482021-01-08T12:24:00+00:00"Zhang, Xuefu"https://zbmath.org/authors/?q=ai:zhang.xiufu"Wu, Shanhe"https://zbmath.org/authors/?q=ai:wu.shanheSummary: Recently, the general width-integral of index \(i\) was introduced and some of its isoperimetric inequalities were established. In this paper, we establish some new Brunn-Minkowski type inequalities for general width-integral of index \(i\).On a Helly-type question for central symmetry.https://zbmath.org/1449.520062021-01-08T12:24:00+00:00"Garber, Alexey"https://zbmath.org/authors/?q=ai:garber.alexei"Roldán-Pensado, Edgardo"https://zbmath.org/authors/?q=ai:roldan-pensado.edgardoIn 2010, K. Swanewpoel posed the following Helly-type problem: determine the existence of \(k\in{\mathbb{N}}\) such that, for any planar set \(X\), if any \(k\) points of \(X\) are in centrally symmetric convex (c.s.c.) position, then the whole set \(X\) is so. Here, it is said that a set is in c.s.c. position if it is contained in the boundary of a centrally symmetric convex body. This question is known to be true for \(k\leq 5\).
In the paper under review, the authors prove that the above problem has not a positive solution when \(k=8\). They moreover show that, if \(X\) is a simple planar closed convex curve, then the answer is affirmative for \(k=6\).
Reviewer: Maria A. Hernández Cifre (Murcia)Asymmetric \({L_p}\)-radial difference bodies.https://zbmath.org/1449.520052021-01-08T12:24:00+00:00"Qi, Jibing"https://zbmath.org/authors/?q=ai:qi.jibingSummary: The notion of asymmetric \({L_p}\)-radial difference bodies about star bodies is defined, and some of their properties are studied. Some inequalities for dual quermassintegrals of asymmetric \({L_p}\)-radial difference bodies are established. In particular, some inequalities for the volumes of asymmetric \({L_p}\)-radial difference bodies are obtained.Curvature in Hilbert geometries.https://zbmath.org/1449.530052021-01-08T12:24:00+00:00"Kurusa, Árpád"https://zbmath.org/authors/?q=ai:kurusa.arpadSummary: We provide more transparent proofs for the facts that the curvature of a Hilbert geometry in the sense of Busemann can not be non-negative and a point of non-positive curvature is a projective center of the Hilbert geometry. Then we prove that the Hilbert geometry has non-positive curvature at its projective centers, and that a Hilbert geometry is a Cayley-Klein model of Bolyai's hyperbolic geometry if and only if it has non-positive curvature at every point of its intersection with a hyperplane. Moreover a 2-dimensional Hilbert geometry is a Cayley-Klein model of Bolyai's hyperbolic geometry if and only if it has two points of non-positive curvature and its boundary is twice differentiable where it is intersected by the line joining those points of non-positive curvature.The isodiametric problem on the sphere and in the hyperbolic space.https://zbmath.org/1449.510252021-01-08T12:24:00+00:00"Böröczky, K. J."https://zbmath.org/authors/?q=ai:boroczky.karoly-jun"Sagmeister, Á."https://zbmath.org/authors/?q=ai:sagmeister.aBy building on ``ideas related to two-point symmetrization'' in [\textit{G. Aubrun}, \textit{M. Fradelizi}, Arch. Math. 82, No. 3, 282--288 (2004; Zbl 1069.52012)], the authors extend to hyperbolic spaces and to the sphere the isodiametric results that were established for Euclidean spaces in [\textit{L. Bieberbach}, Jahresber. Dtsch. Math.-Ver. 24, 247--250 (1915; JFM 45.0623.01)] (for the two-dimensional case) and in [\textit{P. Uryson}, Moscou, Rec. Math. 31, 477--486 (1924; JFM 50.0489.01)]: Measurable and bounded sets of diameter \(\leq D\) (with \(D<\pi\) in the case that the set is on the \(n\)-dimensional sphere of radius \(1\)) have volumes that do not exceed that of a ball of radius \(\frac{D}{2}\), and equality holds if and only if the closure of \(X\) is a ball of radius \(\frac{D}{2}\).
Reviewer: Victor V. Pambuccian (Glendale)The reverse Petty projection inequality.https://zbmath.org/1449.520072021-01-08T12:24:00+00:00"Lin, Youjiang"https://zbmath.org/authors/?q=ai:lin.youjiangSummary: It is proved that if \(K\) is an origin-symmetric convex body in \({\text bf{R}^2}\) and \({\prod ^*}K\) is the polar projection body of \(K\), then the volumes of \(K\) and \({\prod ^*}K\) satisfy the inequality \(V\left(K \right)V\left({{\prod ^*}K} \right) \ge 2\) with equality if \(K\) is a parallelogram.On isoperimetric inequality for mixture of convex and star bodies.https://zbmath.org/1449.520082021-01-08T12:24:00+00:00"Zhao, Changjian"https://zbmath.org/authors/?q=ai:zhao.changjianSummary: In this paper, we establish a new isoperimetric inequality for the mixture of convex and star bodies. Our result in special case yields the classical isoperimetric inequality, which is an improvement and modification of a previous result.Valuation characteristics of linear transformations on \(\mathbb{R}^2\).https://zbmath.org/1449.520092021-01-08T12:24:00+00:00"Wang, Zhenxin"https://zbmath.org/authors/?q=ai:wang.zhenxin"Guo, Qi"https://zbmath.org/authors/?q=ai:guo.qiSummary: In this paper, we study the Minkowski valuations on the set of 1 or 2 dimensional convex bodies compatible with a linear transformation and translations in some sense. We first introduce a kind of compatibility that all linear transformations (as Minkowski valuations) possess naturally. Then, we show that, under some natural conditions, monotone Minkowski valuations with such compatibility are exactly linear transformations. So we obtain a valuation characterization of linear transformations on Euclidean 1 or 2-spaces.Semi-polytope decomposition of a generalized permutohedron.https://zbmath.org/1449.050192021-01-08T12:24:00+00:00"Oh, Suho"https://zbmath.org/authors/?q=ai:oh.suhoSummary: In this short note, we show explicitly how to decompose a generalized permutohedron into semi-polytopes.Characterization of the cross-linked fibrils under axial motion constraints with graphs.https://zbmath.org/1449.530022021-01-08T12:24:00+00:00"Nagy Kem, Gyula"https://zbmath.org/authors/?q=ai:nagy-kem.gyulaSummary: The filament networks play a significant role in biomaterials as structural stability and transmit mechanical signs. Introducing a 3D mechanical model for the infinitesimal motion of cross-linked fibrils under axial motion constraints, we provide a graph theoretical model and give the characterization of the flexibility and the rigidity of this framework. The connectedness of the graph \(G(v,r)\) of the framework in some cases characterizes the flexibility and rigidity of these structures. In this paper, we focus on the kinematical properties of fibrils and proof the next theorem for generic nets of fibrils that are cross-linked by another type of fibrils. ``If the fibrils and the bars are generic positions, the structure will be rigid if and only if each of the components of \(G(v,r)\) has at least one circuit.'' We offer some conclusions, including perspectives and future developments in the frameworks of biostructures as microtubules, collagens, celluloses, actins, other polymer networks, and composite which inspired this work.Bases and subbases in \((L, M)\)-fuzzy convex spaces.https://zbmath.org/1449.520012021-01-08T12:24:00+00:00"Pang, Bin"https://zbmath.org/authors/?q=ai:pang.binSummary: Considering \(L\) being a continuous lattice and \(M\) being a completely distributive lattice, bases and subbases in \((L, M)\)-fuzzy convex spaces are investigated. In an axiomatic approach, axiomatic bases and axiomatic subbases are proposed. It is shown that axiomatic bases and axiomatic subbases can be used to generate \((L, M)\)-fuzzy convex structures and some of their applications are investigated.