B. Klartag
We investigate the rate of convergence in the central limit theorem for convex sets. We obtain bounds with a power-law dependence on the dimension. These bounds are asymptotically better than the logarithmic estimates which follow from the original proof of the central limit theorem for convex sets.
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Peelings and Wrappings of Families of Convex Sets with Applications to Strongly Convex Sets Generated by Random Samples
本文引入并研究紧凸集族的剥皮与包裹操作，建立几何框架，并证明在严格凸正则条件下，随机K-壳的极体剥皮收敛于泊松极限的相应剥皮。
On the Maximal Gaussian Perimeter of Convex Sets, Revisited
本文重新研究纳扎罗夫构造的近乎最大高斯表面积凸集，并利用凸影响概念提供了新的分析方法。
In Guo and Peng's article [Spherically convex sets and spherically convex functions, J. Convex Anal. 28 (2021), 103--122], one defines the notions of spherical convex sets and functions on "general curved surfaces" in $\mathbb{R}^{n}$ $(n\ge2)$, one studies several properties of these classes of sets and functions, and one establishes analogues of Radon, Helly, Carathéodory and Minkowski theorems for spherical convex sets, as well as some properties of spherical convex functions which are analogous to those of usual convex functions. In obtaining such results, the authors use an analytic appro…
The problem of shadow is solved. It is equivalent to condition for point is in generalized convex hull of a family of compact sets.
In a partially ordered semigroup with the duality (or polarity) transform, it is possible to define a generalisation of continued fractions. General sufficient conditions for convergence of continued fractions with deterministic terms are provided. Two particular applications concern the cases of convex sets with the Minkowski addition and the polarity transform (where also necessary and sufficient conditions of convergence for continued fractions with constant terms are obtained) and the family of non-negative convex functions with the Legendre--Fenchel and Artstein-Avidan--Milman transforms.
We study intersections of projective convex sets in the sense of Steinitz. In a projective space, an intersection of a nonempty family of convex sets splits into multiple connected components each of which is a convex set. Hence, such an intersection is called a multi-convex set. We derive a duality for saturated multi-convex sets: there exists an order anti-isomorphism between nonempty saturated multi-convex sets in a real projective space and those in the dual projective space. In discrete geometry and computational geometry, these results allow to transform a given problem into a dual probl…
The work consists of solutions of metric problems for convex and finite subsets of geodesic spaces.
The Legendre Transform of Convex Lattice Sets
本文通过离散勒让德变换研究凸格集，定义其极集并证明自对偶性质，同时建立包含、并交等性质，并揭示交叉多胞体与离散马勒积最小性之间的等价关系。
In this short note we provide a new proof of the recent result of Han and Nashimura on the separation of spherical convex sets established in arXiv:2002.06558. Our proof is based on a result stated in locally convex spaces.
Extending results of Suss and Hadwiger (proved by them for the case of convex bodies and positive ratios), we show that compact (respectively, closed) convex sets in the Euclidean space of dimension n are homothetic provided for any given integer m between 2 and n - 1 (respectively, between 3 and n - 1), the orthogonal projections of the sets on every m-dimensional plane are homothetic, where homothety ratio and its sign may depend on the projection plane. The proof uses a refined version of Straszewicz's theorem on exposed points of compact convex sets.
Minkowski's classical existence theorem provides necessary and sufficient conditions for a Borel measure on the unit sphere of Euclidean space to be the surface area measure of a convex body. The solution is unique up to a translation. We deal with corresponding questions for unbounded convex sets, whose behavior at infinity is determined by a given closed convex cone. We provide an existence theorem and a stability result.
On the Busemann-Petty problem concerning central sections of centrally symmetric convex bodies
我们提出了一种方法，它表明在$\Eb$的Busemann-Petty问题，关于中心对称的凸体的中心部分，有一个积极的答案。与其他结果一起，这解决了每个维度的问题。
There were obtained some properties of weakly m-convex sets. Various kinds of the problem of shadow were investigated. There was obtained the lower estimation for the number of balls that are necessary to create a shadow at the point of the sphere in three-dimensional Euclidean space
In this paper, we study bijections on strictly convex sets of $\mathbf R \mathbf P^n$ for $n \geq 2$ and closed convex projective surfaces equipped with the Hilbert metric that map complete geodesics to complete geodesics as sets. Hyperbolic $n$-space with its standard metric is a special example of the spaces we consider, and it is known that these bijections in this context are precisely the isometries. We first prove that this result generalizes to an arbitrary strictly convex set. For the surfaces setting, we prove the equivalence of mapping simple closed geodesics to simple closed geodesi…
It is considered a special, convex variant of Sperner lemma type .
In this paper, we study the number of compact sets needed in an infinite family of convex sets with a local intersection structure to imply a bound on its piercing number, answering a conjecture of Erdős and Grünbaum. Namely, if in an infinite family of convex sets in $\mathbb{R}^d$ we know that out of every $p$ there are $q$ which are intersecting, we determine if having some compact sets implies a bound on the number of points needed to intersect the whole family. We also study variations of this problem.
Two elements, $x$ and $y$, are separated by a set $S$ if it contains exactly one of $x$ and $y$. We prove that any set of $n$ points in general position in the plane can be separated by $O(n\log\log n/\log n)$ convex sets, and for some point sets $Ω(n/\log n)$ convex sets are necessary.
It is shown that, for any function $g$ that is weakly increasing on compact convex sets and has the property that if $λ\ge 0$ and $K^\prime$ is a translate of $λK$ then $g(K^\prime) =λg(K)$, then for any covering $\bigcup_i X_i\supseteq X$ of a compact convex set $X$ by finitely many compact convex sets $X_i$, the inequality $g(X) \leq \sum_i g(X_i)$ holds. Among the consequences is an affirmative solution of Bang's Plank Problem.
In the setting of step two Carnot groups, we show a "cone property" for horizontally convex sets. Namely we prove that, given a horizontally convex set $C$, a pair of points $P\in \partial C$ and $Q\in $ int $C$, both belonging to a horizontal line $\ell$, then an open truncated subRiemannian cone around $\ell$ and with vertex at $P$ is contained in $C$. We apply our result to the problem of classification of horizontally monotone sets in Carnot groups. We are able to show that monotone sets in the direct product $\mathbb{H} \times\mathbb{R}$ of the Heisenberg group with the real line have hyp…
Aperiodic sets of three types of convex polygons
本研究基于手性非周期单铺块Tile(1, 1)，通过将其内部划分为五个边数≥5的凸多边形，构造出三类边无匹配条件的非周期凸五边形集合，目前已明确四种构造方法。
In this article a class of closed convex sets in the Euclidean $n$-space which are the convex hull of their profiles is described. Thus a generalization of Krein-Milman theorem\cite{Lay:1982} to a class of closed non-compact convex sets is obtained. Sufficient and necessary conditions for convexity, affinity and starshapedness of a closed set and its boundary have been derived in terms of their boundary points.
The famous Hadwiger theorem classifies all rigid motion invariant continuous valuations on convex sets as linear conbinations of quermassintegrals. We prove much more general result. We classify continuous valuations which are invariant with respect to the orthogonal (or special orthogonal) group. Some applications to integral geometry are given.
Let $K$ be a compact convex set and $m$ be a positive integer. The covering functional of $K$ with respect to $m$ is the smallest $λ\in[0,1]$ such that $K$ can be covered by $m$ translates of $λK$. Estimations of the covering functionals of convex hulls of two or more compact convex sets are presented. It is proved that, if a three-dimensional convex body $K$ is the convex hull of two compact convex sets having no interior points, then the least number $c(K)$ of smaller homothetic copies of $K$ needed to cover $K$ is not greater than $8$ and $c(K)=8$ if and only if $K$ is a parallelepiped.
If the n-dimensional unit sphere is covered by finitely many spherically convex bodies, then the sum of the inradii of these bodies is at least π. This bound is sharp, and the equality case is characterized.
Properties of several sorts of lattices of convex subsets of R^n are examined. The lattice of convex sets containing the origin turns out, for n>1, to satisfy a set of identities strictly between those of the lattice of all convex subsets of R^n and the lattice of all convex subsets of R^{n-1}. The lattices of arbitrary, of open bounded, and of compact convex sets in R^n all satisfy the same identities, but the last of these is join-semidistributive, while for n>1 the first two are not. The lattice of relatively convex subsets of a fixed set S \subseteq R^n satisfies some, but in general not a…
We prove that for any non-symmetric irreducible divisible convex set, the proximal limit set is the full projective boundary.
The growth rate of surface area measure for noncompact convex sets with prescribed asymptotic cone
本文研究了一类无界闭凸集的Minkowski问题，将其等价转化为有界凸开域上可能不可积数据的Monge-Ampère方程，并给出了2维情形的完整解及高维情形的部分结果。
The main result implies that a proper convex subset of an irreducible higher rank symmetric space cannot have Zariski dense stabilizer.
Minimal pairs of convex sets which share a recession cone
该文研究具有相同回收锥的无界凸集极小对，证明相关性质，给出二维情形的计算公式与极小性判据，推广多面体和项判据，并展示其在dc-函数极小表示中的应用。
We prove some results concerning the boundary of a convex set in $\H^n$. This includes the convergence of curvature measures under Hausdorff convergence of the sets, the study of normal points, and, for convex surfaces, a generalized Gauss equation and some natural characterizations of the regular part of the Gaussian curvature measure.
Strongly convex sets in Hilbert spaces are characterized by local properties. One quantity which is used for this purpose is a generalization of the modulus of convexity δ_Ωof a set Ω. We also show that \lim_{ε\to 0} δ_Ω(ε)/ε^2 exists whenever Ωis closed and convex.
While faces of a polytope form a well structured lattice, in which faces of each possible dimension are present, this is not true for general compact convex sets. We address the question of what dimensional patterns are possible for the faces of general closed convex sets. We show that for any finite sequence of positive integers there exist compact convex sets which only have extreme points and faces with dimensions from this prescribed sequence. We also discuss another approach to dimensionality, considering the dimension of the union of all faces of the same dimension. We show that the ques…
After surveying some known properties of compact convex sets in the plane, we give a two rigorous proofs of the general feeling that supporting lines can be slide-turned slowly and continuously. Targeting a wide readership, our treatment is elementary on purpose.
Analytic aspects of the dilation inequality for symmetric convex sets in Euclidean spaces
本文探讨欧几里得空间中对称凸集膨胀不等式的解析形式，证明其等价于对称拟凸函数相对熵的界，并由此研究多个相关不等式，通过有界扰动与张量化构造满足该不等式的新概率测度。
We establish the following max-plus analogue of Minkowski's theorem. Any point of a compact max-plus convex subset of $(R\cup\{-\infty\})^n$ can be written as the max-plus convex combination of at most $n+1$ of the extreme points of this subset. We establish related results for closed max-plus convex cones and closed unbounded max-plus convex sets. In particular, we show that a closed max-plus convex set can be decomposed as a max-plus sum of its recession cone and of the max-plus convex hull of its extreme points.
We study existence and partial regularity relative to the weighted Steiner problem in Banach spaces. We show $C^1$ regularity almost everywhere for almost minimizing sets in uniformly rotund Banach spaces whose modulus of uniform convexity verifies a Dini growth condition.
Let K be the symmetric convex hull of m independent random vectors uniformly distributed on the unit sphere of R^n. We prove that, for every $δ>0$, the isotropy constant of K is bounded by a constant $c(δ)$ with high probability, provided that $m\geq (1+δ)n$.
In this paper we focus on the tropical convex hull of convex sets and polyhedral complexes. We give a vertex description of the tropical convex hull of a line segment and a ray. %in \RR^{n+1}/\RR\mathbf{1}. Next we show that tropical convex hull and ordinary convex hull commute in two dimensions and characterize tropically convex polyhedra in any dimension. %$\mathbb{R}^3/\mathbb {R}\mathbf{1}$. Finally we show that the dimension of a tropically convex fan depends on the coordinates of its rays and give a lower bound on the degree of a fan tropical curve using only tropical techniques.
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