K. Bezdek and Gy. Kiss showed that existence of origin-symmetric coverings of unit sphere in $\mathbb{E}^n$ by at most $2^n$ congruent spherical caps with radius not exceeding $\arccos\sqrt{\frac{n-1}{2n}}$ implies the $X$-ray conjecture and the illumination conjecture for convex bodies of constant width in $\mathbb{E}^n$, and constructed such coverings for $4\le n\le 6$. Here we give such constructions with fewer than $2^n$ caps for $5\le n\le 15$.
For the illumination number of any convex body of constant width in $\mathbb{E}^n$, O.~Schramm proved an upper estimate with exponential growth of order $(3/2)^{n/2}$. In particular, that estimate is less than $3\cdot 2^{n-2}$ for $n\ge 16$, confirming the above mentioned conjectures for the class of convex bodies of constant width. Thus, our result settles the outstanding cases $7\le n\le 15$.
We also show how to calculate the covering radius of a given discrete point set on the sphere efficiently on a computer.
Covering numbers of convex bodies based on homothetical copies and related illumination numbers are well-known in combinatorial geometry and, for example, related to Hadwiger's famous covering problem. Similar numbers can be defined by using proper translates instead of homothets, and even more related concepts make sense. On these lines we introduce some new covering and illumination numbers of convex bodies, present their properties and compare them with each other as well as with already known numbers. Finally, some suggestive examples illustrate that these new illumination numbers are inte…
This paper suveys different variants of the following problem: Given a convex set $K$ and a sequence $\{C_i\}$ of convex bodies in $E^n$, is it possible to pack the sequence of bodies in $K$ or cover $K$ with the bodies? Algorithmic versions of these problems are on-line packing and on-line covering: The bodies of the sequence are given one at a time and the algorithm is to decide on the placement of the arriving body before the next body is revealed; once placed, the body cannot be moved.
Covering a convex body by its homothets is a classical notion in discrete geometry that has resulted in a number of interesting and long-standing problems. Swanepoel introduced the covering parameter of a convex body as a means of quantifying its covering properties. In this paper, we introduce two relatives of the covering parameter called covering index and weak covering index, which upper bound well-studied quantities like the illumination number, the illumination parameter and the covering parameter of a convex body. Intuitively, the two indices measure how well a convex body can be covere…
In connection with an unsolved problem of Bang (1951) we give a lower bound for the sum of the base volumes of cylinders covering a d-dimensional convex body in terms of the relevant basic measures of the given convex body. As an application we establish lower bounds on the number of k-dimensional flats (i.e. translates of k-dimensional linear subspaces) needed to cover all the integer points of a given convex body in d-dimensional Euclidean space for 0<k<d.
The paper is devoted to coverings by translative homothets and illuminations of convex bodies. For a given positive number $α$ and a convex body $B$, $g_α(B)$ is the infimum of $α$-powers of finitely many homothety coefficients less than 1 such that there is a covering of $B$ by translative homothets with these coefficients. $h_α(B)$ is the minimal number of directions such that the boundary of $B$ can be illuminated by this number of directions except for a subset whose Hausdorff dimension is less than $α$. In this paper, we prove that $g_α(B)\leq h_α(B)$, find upper and lower bounds for both…
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.
This article gives estimates on covering numbers and diameters of random proportional sections and projections of symmetric quasi-convex bodies in $\mathbb R$. These results were known for the convex case and played an essential role in development of the theory. Because duality relations can not be applied in the quasi-convex setting, new ingredients were introduced that give new understanding for the convex case as well.
In geometry, there are several challenging problems studying numbers associated to convex bodies. For example, the packing density problem, the kissing number problem, the covering density problem, the packing-covering constant problem, Hadwiger's covering conjecture and Borsuk's partition conjecture. They are fundamental and fascinating problems about the same objects. However, up to now, both the methodology and the technique applied to them are essentially different. Therefore, a common foundation for them has been much expected. By treating problems of these types as functionals defined on…
In [BL] in relation to the unsolved Bang's plank problem (1951) we obtained a lower bound for the sum of relevant measures of cylinders covering a given d-dimensional convex body. In this paper we provide the packing counterpart of these estimates. We also extend bounds to the case of r-fold covering and packing and show a packing analog of Falconer's results ([Fa]).
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…
Let $\| \cdot \|$ be the euclidean norm on ${\bf R}^n$ and $γ_n$ the (standard) Gaussian measure on ${\bf R}^n$ with density $(2 π)^{-n/2} e^{- \| x\|^2 /2}$. Let $\vartheta$ ($ \simeq 1.3489795$) be defined by $γ_1 ([ - \vartheta /2, \vartheta /2]) = 1/2$ and let $L$ be a lattice in ${\bf R}^n$ generated by vectors of norm $\leq \vartheta$. Then, for any closed convex set $V$ in ${\bf R}^n$ with $γ_n (V) \geq \frac{1}{2}$ and for any $a \in {\bf R}^n$, $(a +L) \cap V \neq ϕ$. The above statement can be viewed as a ``nonsymmetric'' version of Minkowski Theorem.
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 …
Nearly Sharp Bounds for Lattice Coverings by Convex Bodies
School of Mathematics, Shandong University（山东大学数学学院）；Extremal Combinatorics and Probability Group, Institute for Basic Science（基础科学研究所极值组合与概率组）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）
In this article we encode Hadwiger's covering conjecture and Borsuk's partition conjecture into continuous functions defined on the spaces of convex bodies, propose a four-step program to approach them, and obtain some partial results.
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…
We consider the Tarski--Bang problem about covering of convex bodies by planks. The results of this kind give a lower bound on the sum of widths of planks (regions between a pair of parallel hyperplanes) covering a given convex body. Previously we have applied some notions of symplectic geometry to study convex bodies, and here we show that the symplectic techniques may be useful in this problem as well. We are able to handle some particular cases with the symplectic techniques, and show that the general cases would follow from a certain ``subadditivity conjecture'' in symplectic geometry, mot…
Let $K$ be a $d$ dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote by $K_n$ the convex hull of $n$ points chosen randomly and independently from $K$ according to the uniform distribution. Matching lower and upper bounds are obtained for the orders of magnitude of the variances of the $s$-th intrinsic volumes $V_s(K_n)$ of $K_n$ for $s\in\{1, ..., d\}$. Furthermore, strong laws of large numbers are proved for the intrinsic volumes of $K_n$. The essential tools are the Economic Cap Covering Theorem of Bárány and La…
Motivated by long-standing conjectures on the discretization of classical inequalities in the Geometry of Numbers, we investigate a new set of parameters, which we call \emph{packing minima}, associated to a convex body $K$ and a lattice $Λ$. These numbers interpolate between the successive minima of $K$ and the inverse of the successive minima of the polar body of $K$, and can be understood as packing counterparts to the covering minima of Kannan & Lovász (1988). As our main results, we prove sharp inequalities that relate the volume and the number of lattice points in $K$ to the sequence of …
A central problem in discrete geometry, known as Hadwiger's covering problem, asks what the smallest natural number $N\left(n\right)$ is such that every convex body in ${\mathbb R}^{n}$ can be covered by a union of the interiors of at most $N\left(n\right)$ of its translates. Despite continuous efforts, the best general upper bound known for this number remains as it was more than sixty years ago, of the order of ${2n \choose n}n\ln n$. In this note, we improve this bound by a sub-exponential factor. That is, we prove a bound of the order of ${2n \choose n}e^{-c\sqrt{n}}$ for some universal co…
For two convex bodies K and T in $R^n$, the covering number of K by T, denoted N(K,T), is defined as the minimal number of translates of T needed to cover K. Let us denote by $K^o$ the polar body of K and by D the euclidean unit ball in $R^n$. We prove that the two functions of t, N(K, tD) and N(D, tK^o), are equivalent in the appropriate sense, uniformly over symmetric convex bodies K in $R^n$ and over positive integers n. In particular, this verifies the duality conjecture for entropy numbers of linear operators, posed by Pietsch in 1972, in the central case when either the domain or the ran…
A 1972 duality conjecture due to Pietsch asserts that the entropy numbers of a compact operator acting between two Banach spaces and those of its adjoint are (in an appropriate sense) equivalent. This is equivalent to a dimension free inequality relating covering (or packing) numbers for convex bodies to those of their polars. The duality conjecture has been recently proved (see math.FA/0407236) in the central case when one of the Banach spaces is Hilbertian, which - in the geometric setting - corresponds to a duality result for symmetric convex bodies in Euclidean spaces. In the present paper…
In 1978, Makai Jr. established a remarkable connection between the volume-product of a convex body, its maximal lattice packing density and the minimal density of a lattice arrangement of its polar body intersecting every affine hyperplane. Consequently, he formulated a conjecture that can be seen as a dual analog of Minkowski's fundamental theorem, and which is strongly linked to the well-known Mahler-conjecture. Based on the covering minima of Kannan & Lovász and a problem posed by Fejes Tóth, we arrange Makai Jr.'s conjecture into a wider context and investigate densities of lattice arrange…
Let $C\subset {\mathbb R}^n$ be a convex body. We introduce two notions of convexity associated to C. A set $K$ is $C$-ball convex if it is the intersection of translates of $C$, or it is either $\emptyset$, or ${\mathbb R}^n$. The $C$-ball convex hull of two points is called a $C$-spindle. $K$ is $C$-spindle convex if it contains the $C$-spindle of any pair of its points. We investigate how some fundamental properties of conventional convex sets can be adapted to $C$-spindle convex and $C$-ball convex sets. We study separation properties and Carathéodory numbers of these two convexity structu…
If a convex body $K \subset \mathbb{R}^n$ is covered by the union of convex bodies $C_1, \ldots, C_N$, multiple subadditivity questions can be asked. Two classical results regard the subadditivity of the width (the smallest distance between two parallel hyperplanes that sandwich $K$) and the inradius (the largest radius of a ball contained in $K$): the sum of the widths of the $C_i$ is at least the width of $K$ (this is the plank theorem of Thoger Bang), and the sum of the inradii of the $C_i$ is at least the inradius of $K$ (this is due to Vladimir Kadets). We adapt the existing proofs of the…
