Rafik Aramyan
The article considers the problem of existence and uniqueness of centrally symmetrical convex body for which the projection curvature radius function coincides with a given flag function. A necessary and sufficient condition is found that ensures a positive answer. An algorithm for construction the body in question is proposed
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We present a method which shows that in $\Eb$ the Busemann-Petty problem, concerning central sections of centrally symmetric convex bodies, has a positive answer. Together with other results, this settles the problem in each dimension.
In this note, we derive an asymptotically sharp upper bound on the number of lattice points in terms of the volume of centrally symmetric convex bodies. Our main tool is a generalization of a result of Davenport that bounds the number of lattice points in terms of volumes of suitable projections.
In this paper we study the bisections of a centrally symmetric planar convex body which minimize the maximum relative diameter functional. We give necessary and sufficient conditions for being a minimizing bisection, as well as analyzing the behavior of the so-called standard bisection.
Homothetic packings of centrally symmetric convex bodies
本文研究中心对称凸体的位似堆积问题，证明随机半径位似堆积的接触图具有(2,2)-稀疏性，并证明存在余贫集使得任意(2,2)-稀疏平面图可实现为无应力位似堆积的接触图。
On the illumination of centrally symmetric cap bodies in small dimensions
本文研究了小维空间中中心对称帽体的照明数，证明了在三维空间中此类帽体的照明数不超过6，在四维空间中不超过8。
A characterization of centrally symmetric convex bodies in terms of visual cones
本文研究了通过视觉锥特征刻画中心对称凸体与超曲面的几何性质，证明了在特定条件下凸体与超曲面的对称性。
In this paper, we obtain the best possible value of the absolute constant $C$ such that for every isotropic convex body $K \subseteq \mathbb{R}^n$ the following inequality (which was proved by Klartag and reduces the hyperplane conjecture to centrally symmetric convex bodies) is satisfied: $$ L_K\leq CL_{K_{n+2}(g_K)}. $$ Here $L_K$ denotes the isotropic constant of $K$, $g_K$ its covariogram function, which is log-concave, and, for any log-concave function $g$, $K_{n+2}(g)$ is a convex body associated to the log-concave function $g$, which belongs to a uniparametric family introduced by Ball.…
The convex hull of a ball with an exterior point is called a spike (or cap). A union of finitely many spikes of a ball is called a spiky ball. If a spiky ball is convex, then we call it a cap body. In this note we upper bound the illumination numbers of $2$-illuminable spiky balls as well as centrally symmetric cap bodies. In particular, we prove the Illumination Conjecture for centrally symmetric cap bodies in sufficiently large dimensions by showing that any $d$-dimensional centrally symmetric cap body can be illuminated by $<2^d$ directions in Euclidean $d$-space for all $d\geq 20$. Further…
An isoperimetric problem for Fourier zeros of centrally symmetric convex bodies
本文否证了关于中心对称凸体傅里叶零点到原点最大距离由球取得的猜想，证明平面中正十二边形等多边形更优，高维无最大值。
Let $d \ge 2$, and let $K \subset {\Bbb{R}}^d$ be a convex body containing the origin $0$ in its interior. In a previous paper we have proved the following. The body $K$ is $0$-symmetric if and only if the following holds. For each $ω\in S^{d-1}$, we have that the $(d-1)$-volume of the intersection of $K$ and an arbitrary hyperplane, with normal $ω$, attains its maximum if the hyperplane contains $0$. An analogous theorem, for $1$-dimensional sections and $1$-volumes, has been proved long ago by Hammer (\cite{H}). In this paper we deal with the ($(d-2)$-dimensional) surface area, or with lower…
We investigate a duality relation between floating and illumination bodies. The definitions of these two bodies suggest that the polar of the floating body should be similar to the illumination body of the polar. We consider this question for the class of centrally symmetric convex bodies. We provide precise estimates for $(B_p^n)$ and for centrally symmetric convex bodies with everywhere positive Gauss curvature. Our estimates show that equality of the polar of the floating body and the illumination body of the polar can only be achieved in the case of ellipsoids.
Convex bodies with centrally symmetric sections
本文研究Barker-Larman猜想的一个特例，证明当凸体K是中心对称且严格凸，且球B满足一定条件时，K必为椭球。
A Minkowski class is a closed subset of the space of convex bodies in Euclidean space Rn which is closed under Minkowski addition and non-negative dilatations. A convex body in Rn is universal if the expansion of its support function in spherical harmonics contains non-zero harmonics of all orders. If K is universal, then a dense class of convex bodies M has the following property. There exist convex bodies T1; T2 such that M + T1 = T2, and T1; T2 belong to the rotation invariant Minkowski class generated by K. It is shown that every convex body K which is not centrally symmetric has a linear …
It is proved that every convex body in the plane has a point such that the union of the body and its image under reflection in the point is convex. If the body is not centrally symmetric, then it has, in fact, three affinely independent points with this property.
A classical theorem of Alon and Milman states that any $d$ dimensional centrally symmetric convex body has a projection of dimension $m\geq e^{c\sqrt{\ln{d}}}$ which is either close to the $m$-dimensional Euclidean ball or to the $m$-dimensional cross-polytope. We extended this result to non-symmetric convex bodies.
A generalization of pairwise intersecting Minkowski arrangement of centrally symmetric convex bodies is the pairwise intersecting Minkowski arrangement of order $μ$. Here, the homothetic copies of a centrally symmetric convex body are so that none of their interiors intersect the $μ$-kernel of any other. We give general upper and lower bounds on the cardinality of such arrangements, and study two special cases: For $d$-dimensional translates in classical pairwise intersecting Minkowski arrangement we prove that the sharp upper bound is $3^d$. For $μ=1$ the general version yields to another kno…
Borsuk number for planar convex bodies
本文利用图论工具刻画了Borsuk数等于2的平面凸体，给出了Borsuk数为3的实例，并证明欧氏球是唯一Borsuk数为3的中心对称平面凸体。
For a convex body $K\subset\R^n$ and $i\in\{1,...,n-1\}$, the function assigning to any $i$-dimensional subspace $L$ of $\R^n$, the $i$-dimensional volume of the orthogonal projection of $K$ to $L$, is called the $i$-th projection function of $K$. Let $K, K_0\subset \R^n$ be smooth convex bodies of class $C^2_+$, and let $K_0$ be centrally symmetric. Excluding two exceptional cases, that of $(i,j)=(1,n-1)$ and $(i,j)=(n-2,n-1)$, we prove that $K$ and $K_0$ are homothetic if they have two proportional projection functions. The special case when $K_0$ is a Euclidean ball provides an extension of…
Minimal Covering Bodies and a Minkowski-Type Criterion for Lattice Coverings
引入最小覆盖体概念，建立三维格覆盖的构造性准则，证明高维空间中最小覆盖体有无穷多种组合类型，并提出闵可夫斯基型几何准则与代数交框架。
We prove several estimates for the moments of arbitrary measures on convex bodies. We apply these estimates to show a new slicing inequality for measures on convex bodies. We also deduce estimates for the outer volume ratio distance from an arbitrary centrally-symmetric convex body in R^n to the class of unit balls of n-dimensional subspaces of L_p-spaces. Finally, we prove a result of the Busemann-Petty type for these moments.
The spherical centroid body of a centrally-symmetric convex body in the Euclidean unit sphere is introduced. Two alternative definitions - one geometric, the other probabilistic in nature - are given and shown to lead to the same objects. The geometric approach is then used to establish a number of basic properties of spherical centroid bodies, while the probabilistic approach inspires the proof of a spherical analogue of the classical polar Busemann-Petty centroid inequality.
In this note we link symplectic and convex geometry by relating two seemingly different open conjectures: a symplectic isoperimetric-type inequality for convex domains, and Mahler's conjecture on the volume product of centrally symmetric convex bodies. More precisely, we show that if for convex bodies of fixed volume in the classical phase space the Hofer-Zehnder capacity is maximized by the Euclidean ball, then a hypercube is a minimizer for the volume product among centrally symmetric convex bodies.
We study upper bounds on the number of lattice points for convex bodies having their centroid at the origin. For the family of simplices as well as in the planar case we obtain best possible results. For arbitrary convex bodies we provide an upper bound, which extends the centrally symmetric case and which, in particular, shows that the centroid assumption is indeed much more restrictive than an assumption on the number of interior lattice points even for the class of lattice polytopes.
We prove several estimates for the volume, mean width, and the value of the Wills functional of sections of convex bodies in John's position, as well as for their polar bodies. These estimates extend some well-known results for convex bodies in John's position to the case of lower-dimensional sections, which had mainly been studied for the cube and the regular simplex. Some estimates for centrally symmetric convex bodies in minimal surface area position are also obtained.
We consider the question how well a floating body can be approximated by the polar of the illumination body of the polar. We establish precise convergence results in the case of centrally symmetric polytopes. This leads to a new affine invariant which is related to the cone measure of the polytope.
We define new natural variants of the notions of weighted covering and separation numbers and discuss them in detail. We prove a strong duality relation between weighted covering and separation numbers and prove a few relations between the classical and weighted covering numbers, some of which hold true without convexity assumptions and for general metric spaces. As a consequence, together with some volume bounds that we discuss, we provide a bound for the famous Levi-Hadwiger problem concerning covering a convex body by homothetic slightly smaller copies of itself, in the case of centrally sy…
The mathematical theory of rigidity of body-bar and body-hinge frameworks provides a useful tool for analyzing the rigidity and flexibility of many articulated structures appearing in engineering, robotics and biochemistry. In this paper we develop a symmetric extension of this theory which permits a rigidity analysis of body-bar and body-hinge structures with point group symmetries. The infinitesimal rigidity of body-bar frameworks can naturally be formulated in the language of the exterior (or Grassmann) algebra. Using this algebraic formulation, we derive symmetry-adapted rigidity matrices …
High proved the following theorem. If the intersections of any two congruent copies of a plane convex body are centrally symmetric, then this body is a circle. In our paper we extend the theorem of High to spherical and hyperbolic planes. If in any of these planes, or in ${\Bbb R}^2$, there is a pair of closed convex sets with interior points, and the intersections of any congruent copies of these sets are centrally symmetric, then, under some mild hypotheses, our sets are congruent circles, or, for ${\Bbb R}^2$, two parallel strips. We prove the analogue of this statement, for $S^d$, ${\Bbb R…
Flag measures are descriptors of convex bodies $K$ in $d$-dimensional Euclidean space generalizing the classical area measures. They have been used to provide general integral formulas for mixed volumes (see Hug, Rataj and Weil (2017)). Here, we consider an image measure $γ_j(K,\cdot)$ of flag measures, defined on the Grassmannian $G(d,j)$ of affine $j$-spaces, $1\le j\le d-1$, and show that it determines centrally symmetric bodies $K$ of dimension $\geq j+1$ uniquely. We then explain that Grassmann measures appear in the representation of smooth, translation invariant, continuous and even val…
We improve the estimates for the Ekeland--Hofer--Zehnder capacity of convex bodies by Gluskin and Ostrover. In the course of our argument we show that a closed characteristic of minimal action on the boundary of a centrally symmetric convex body in $\mathbb R^{2n}$ must itself be centrally symmetric, with generalizations to some other types of symmetry.
In 1969, Vic Klee asked whether a convex body is uniquely determined (up to translation and reflection in the origin) by its inner section function, the function giving for each direction the maximal area of sections of the body by hyperplanes orthogonal to that direction. We answer this question in the negative by constructing two infinitely smooth convex bodies of revolution about the $x_n$-axis in $\R^n$, $n\ge 3$, one origin symmetric and the other not centrally symmetric, with the same inner section function. Moreover, the pair of bodies can be arbitrarily close to the unit ball.
We introduce the vertex index, vein(K), of a given centrally symmetric convex body K, which, in a sense, measures how well K can be inscribed into a convex polytope with small number of vertices. This index is closely connected to the illumination parameter of a body, introduced earlier by the first named author, and, thus, related to the famous conjecture in Convex Geometry about covering of a d-dimensional body by 2^d smaller positively homothetic copies. We provide asymptotically sharp estimates (up to a logarithmic term) of this index in the general case. Also, we provide sharp estimates i…
Symplectically self-polar polytopes of minimal capacity
本文研究了辛自极化凸体的最小容量，并证明了相关下界不可改进，同时进行了体积相关实验和推测。
The Faber-Krahn position of convex bodies and Gaussian measure inequalities
研究凸体处于法伯 - 克拉恩位置（使第一个狄利克雷特征值在保体积线性轨道内最小化）的唯一性，通过正定线性变形下特征值的对数凸性等方法证明，还得到相关定理新证及其他凸性结果，并改进了中心对称凸体相关不等式界。
In this note we study the expected value of certain symplectic capacities of randomly rotated centrally symmetric convex bodies in the classical phase space.
A stability version of the Blaschke-Santaló inequality and the affine isoperimetric inequality for convex bodies of dimension n>2 is proved. The first step is the reduction to the case when the convex body is o-symmetric and has axial rotational symmetry. This step works for related inequalities compatible with Steiner symmetrization. Secondly, for these convex bodies, a stability version of the characterization of ellipsoids by the fact that each hyperplane section is centrally symmetric is established.
For a convex body $K\subset\R^n$, the $k$th projection function of $K$ assigns to any $k$-dimensional linear subspace of $\R^n$ the $k$-volume of the orthogonal projection of $K$ to that subspace. Let $K$ and $K_0$ be convex bodies in $\R^n$, and let $K_0$ be centrally symmetric and satisfy a weak regularity and curvature condition (which includes all $K_0$ with $\f K_0$ of class $C^2$ with positive radii of curvature). Assume that $K$ and $K_0$ have proportional 1st projection functions (i.e., width functions) and proportional $k$th projection functions. For $2\le k<(n+1)/2$ and for $k=3, n=5…
A translation body of a convex body is the convex hull of two of its translates intersecting each other. In the 1950s, Rogers and Shephard found the extremal values, over the family of $n$-dimensional convex bodies, of the maximal volume of the translation bodies of a given convex body. In our paper, we introduce a normed version of this problem, and for the planar case, determine the corresponding quantities, with one exception, for the four types of volumes regularly used in the literature: Busemann, Holmes-Thompson, and Gromov's mass and mass*. We examine the problem also for higher dimensi…
G. Fejes Tóth posed the following problem: Determine the infimum of the densities of the lattices of closed balls in $\bR^n$ such that each affine $k$-subspace $(0 \le k \le n-1)$ of $\bR^n$ intersects some ball of the lattice. We give a lower estimate for any $n,k$ like above. If, in the problem posed by G. Fejes Tóth, we replace the ball $B^n$ by a (centrally symmetric) convex body $K\subset \bR^n$, we may ask for the infimum of all above infima of densities of lattices of translates of $K$ with the above property, when $K$ ranges over all (centrally symmetric) convex bodies in $\bR^n$. For …
Illumination number of 3-dimensional cap bodies
本文证明了3维帽体的照明数为6，验证了该类别的照明猜想。
We provide general estimates which compare the quermassintegrals of a convex body $K$ in ${\mathbb R}^n$ with the averages of the corresponding quermassintegrals of the $k$-codimensional sections of $K$ over $G_{n,n-k}$. An example is the inequality $$α_{n,k,j}\frac{W_j(K)}{|K|}\leq\int_{G_{n,n-k}}\frac{W_j(K\cap F)}{|K\cap F|}dν_{n,n-k}(F)\leq β_{n,k,j}\frac{W_j(K)}{|K|}$$ where the constants $α_{n,k,j}$ and $β_{n,k,j}$ depend only on $n,k$ and $j$, which holds true for any centrally symmetric convex body $K$ in ${\mathbb R}^n$ and any $0\leq j\leq n-k-1\leq n-1$. Using these estimates we obt…
In this paper, the isodiametric problem for centrally symmetric convex bodies in the Euclidean d-space R^d containing no interior non-zero point of a lattice L is studied. It is shown that the intersection of a suitable ball with the Dirichlet-Voronoi cell of 2L is extremal, i.e., it has minimum diameter among all bodies with the same volume. It is conjectured that these sets are the only extremal bodies, which is proved for all three dimensional and several prominent lattices.
We study the volume ratio between 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 $$ \mbox{vr}(PK, QL) \geq c \, \min\left\{\frac{ k}{ \sqrt{n}} \, \sqrt{\frac{1}{\log \log…
Ball characterizations in planes and spaces of constant curvature, I
本文在常曲率平面和空间中，通过任意全等拷贝的交集或并集凸包的对称性条件，刻画了圆、球、平行带及球面型边界分支等几何体。
This paper contains a number of results related to volumes of projective perturbations of convex bodies and the Laplace transform on convex cones. First, it is shown that a sharp version of Bourgain's slicing conjecture implies the Mahler conjecture for convex bodies that are not necessarily centrally-symmetric. Second, we find that by slightly translating the polar of a centered convex body, we may obtain another body with a bounded isotropic constant. Third, we provide a counter-example to a conjecture by Kuperberg on the distribution of volume in a body and in its polar.
A sharp Santaló inequality in a slab with centrally symmetric sections
本文证明了一个尖锐的Santaló型不等式，通过射影对应转化为板条中凸体的体积乘积上界问题，并分类了极值情形，同时给出了一维函数不等式族。
Given a centrally symmetric convex body $K \subset \mathbb{R}^d$ and a positive number $λ$, we consider, among all ellipsoids $E \subset \mathbb{R}^d$ of volume $λ$, those that best approximate $K$ with respect to the symmetric difference metric, or equivalently that maximize the volume of $E\cap K$: these are the maximal intersection (MI) ellipsoids introduced by Artstein-Avidan and Katzin. The question of uniqueness of MI ellipsoids (under the obviously necessary assumption that $λ$ is between the volumes of the John and the Loewner ellipsoids of $K$) is open in general. We provide a positiv…
On a Gallai-type problem and illumination of spiky balls and cap bodies
本文研究两两边交球族的 Gallai 型刺穿问题，并将结果用于刺球与帽体照明，显著改进高维上下界。
On Axial Symmetry in Convex Bodies
研究凸体的轴对称性度量，证明平面凸体的轴对称性下界为约0.69476，并发现轴对称性度量的改进界及高维凸体的推广。
Let K \subset R^N be a convex body containing the origin. A measurable set G \subset R^N with positive Lebesgue measure is said to be uniformly K-dense if, for any fixed r > 0, the measure of G \cap (x + rK) is constant when x varies on the boundary of G (here, x + rK denotes a translation of a dilation of K). We first prove that G must always be strictly convex and at least C1,1-regular; also, if K is centrally symmetric, K must be strictly convex, C1,1-regular and such that K = G - G up to homotheties; this implies in turn that G must be C2,1- regular. Then for N = 2, we prove that G is unif…
For each i = 1, ..., n constructions are given for convex bodies K and L in n-dimensional Euclidean space such that each rank i orthogonal projection of K can be translated inside the corresponding projection of L, even though K has strictly larger m-th intrinsic volumes (i.e. V_m(K) > V_m(L)) for all m > i. It is then shown that, for each i = 1, ..., n, there is a class of bodies C{n,i}, called i-cylinder bodies of R^n, such that, if the body L with i-dimensional covering shadows is an i-cylinder body, then K will have smaller n-volume than L. The families C{n,i} are shown to form a strictly …
Let L be a compact convex set in R^n, and let 1 <= d <= n-1. The set L is defined to be d-decomposable if L is a direct Minkowski sum (affine Cartesian product) of two or more convex bodies each of dimension at most d. A compact convex set L is called d-reliable if, whenever each d-dimensional orthogonal projection of L contains a translate of the corresponding d-dimensional projection of a compact convex set K, it must follow that L contains a translate of K. It is shown that, for 1 <= d <= n-1: (1) d-decomposability implies d-reliability. (2) A compact convex set L in R^n is d-reliable if an…
Mean distance between points inside and on the boundary of a convex body
本文研究扎波罗热茨-塔拉索夫猜想，针对中心对称凸体、平面凸体、外切凸体等不同情况，通过构造反例、推导公式等方式取得相关结论，还证明该猜想对三角形的逐投影形式成立。
The generalized Busemann-Petty problem asks whether centrally-symmetric convex bodies having larger volume of all m-dimensional sections necessarily have larger volume. When m>3 this is known to be false, but the cases m=2,3 are still open. In those cases, it is shown that when the smaller body's radial function is a (n-m)-th root of the radial function of a convex body, the answer to the generalized Busemann-Petty problem is positive (for any larger star-body). Several immediate corollaries of this observation are also discussed.
A subset of a convex body $B$ containing the origin in a Euclidean space is {\it parkable in $B$} if it can be translated inside $B$ in such a manner that the translate the origin. We provide characterizations of ellipsoids and of centrally symmetric convex bodies in Euclidean spaces of dimension $\ge 3$ based on the notion of parkability, answering several questions posed by G. Bergman. The techniques used, which are based on characterizations of Hilbert spaces among finite-dimensional Banach spaces in terms of their lattices of subspaces and algebras of endomorphisms, also apply to improve a…
Ball characterizations in planes and spaces of constant curvature, II \vskip.1cm \centerline{\rm{This pdf-file is not identical with the printed paper.}}
本文推广 High 定理至常曲率空间，刻画了交集对称或并集凸包对称的凸体，证明其为球或圆，并给出双曲平面上的九种情形。
Distances between non-symmetric convex bodies: optimal bounds up to polylog
本文确定了无对称假设的n维凸体Banach-Mazur紧集直径的多对数因子最优上界，并研究了部分包含距离，证明其具有对数上界，核心依赖于各向同性对数凹测度与高斯测度的规范序双向比较。
Let $C$ and $K$ be centrally symmetric convex bodies in ${\mathbb R}^n$. We show that if $C$ is isotropic then \begin{equation*}\|{\bf t}\|_{C^s,K}=\int_{C}\cdots\int_{C}\Big\|\sum_{j=1}^st_jx_j\Big\|_K\,dx_1\cdots dx_s \leq c_1L_C(\log n)^5\,\sqrt{n}M(K)\|{\bf t}\|_2\end{equation*} for every $s\geq 1$ and ${\bf t}=(t_1,\ldots ,t_s)\in {\mathbb R}^s$, where $L_C$ is the isotropic constant of $C$ and $M(K):=\int_{S^{n-1}}\|ξ\|_Kdσ(ξ)$. This reduces a question of V.~Milman to the problem of estimating from above the parameter $M(K)$ of an isotropic convex body. The proof is based on an observati…
The Gaussian measure of a convex body controls its maximal covering radius
本文证明凸体 $V$ 的高斯测度可统一控制 Euclidean 球情形下的向量平衡常数及其格对应量，解决了 Banaszczyk–Szarek 猜想并去除了中心对称限制。
The Fermat-Weber center of a planar body $Q$ is a point in the plane from which the average distance to the points in $Q$ is minimal. We first show that for any convex body $Q$ in the plane, the average distance from the Fermat-Weber center of $Q$ to the points of $Q$ is larger than ${1/6} \cdot Δ(Q)$, where $Δ(Q)$ is the diameter of $Q$. This proves a conjecture of Carmi, Har-Peled and Katz. From the other direction, we prove that the same average distance is at most $\frac{2(4-\sqrt3)}{13} \cdot Δ(Q) < 0.3490 \cdot Δ(Q)$. The new bound substantially improves the previous bound of $\frac{2}{3…
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