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 packing minima. Moreover, we extend classical transference bounds and discuss a natural class of examples in detail.
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.
We study the contact structure of totally separable} packings of translates of a convex body $K$ in $\mathbb{R}^d$, that is, packings where any two touching bodies have a separating hyperplane that does not intersect the interior of any translate in the packing. The separable Hadwiger number $H_{\text{sep}}(K)$ of $K$ is defined to be the maximum number of translates touched by a single translate, with the maximum taken over all totally separable packings of translates of $K$. We show that for each $d\geq 8$, there exists a smooth and strictly convex $K$ in $\mathbb{R}^d$ with $H_{\text{sep}}(…
We consider a typical (in the sense of Baire categories) convex body $K$ in $\mathbb{R}^{d+1}$. The set of feet of its double normals is a Cantor set, having lower box-counting dimension $0$ and packing dimension $d$. The set of lengths of those double normals is also a Cantor set of lower box-counting dimension $0$. Its packing dimension is equal to $\frac{1}{2}$ if $d=1$, is at least $\frac{3}{4}$ if $d=2$, and equals $1$ if $d\geq3$. We also consider the lower and upper curvatures at feet of double normals of $K$, with a special interest for local maxima of the length function (they are cou…
We give the sharp lower bound of the volume product of three dimensional convex bodies which are invariant under a discrete subgroup of $O(3)$ in several cases. We also characterize the convex bodies with the minimal volume product in each case. In particular, this provides a new partial result of the non-symmetric version of Mahler's conjecture in the three dimensional case.
In an old paper of the author the thinnest five-neighbour packing of translates of a convex disc (different from a parallelogram) was determined. The minimal density was $3/7$, and was attained for a certain packing of triangles. In that paper it was announced that for centrally symmetric convex plates (different from a parallelogram) the analogous minimal density was $9/14$, and was attained for a certain packing of affine regular hexagons, and a very sketchy idea of the proof was given. In this paper we give details of this proof.
We study inequalities that simultaneously relate the number of lattice points, the volume and the successive minima of a convex body to one another. One main ingredient in order to establish these relations is Blaschke's shaking procedure, by which the problem can be reduced from arbitrary convex bodies to anti-blocking bodies. As a consequence of our results, we obtain an upper bound on the lattice point enumerator in terms of the successive minima, which is equivalent to Minkowski's upper bound on the volume in terms of the successive minima.
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 give the sharp lower bound of the volume product of $n$-dimensional convex bodies which are invariant under a discrete subgroup $SO(K)=\{ g \in SO(n); g(K)=K \}$, where $K$ is an $n$-cube or $n$-simplex. This provides new partial results of Mahler's conjecture and its non-symmetric version. In addition, we give partial answers for Viterbo's isoperimetric type conjecture in symplectic geometry from the view point of Mahler's conjecture.
We show that a jammed packing of disks with generic radii, in a generic container, is such that the minimal number of contacts occurs and there is only one dimension of equilibrium stresses. We also point out some connections to packings with different radii and results in the theory of circle packings whose graph forms a triangulation of a given topological surface. We also point out a counterexample, due to F. Nazarov, to a previous conjecture that that triangulated packings with fixed numbers of disks with fixed numbers of disks for each radius claiming that such packings were the most dens…
A packing of translates of a convex body in the $d$-dimensional Euclidean space $\mathbb{E}^d$ is said to be totally separable if any two packing elements can be separated by a hyperplane of $\mathbb{E}^{d}$ disjoint from the interior of every packing element. We call the packing $\mathcal P$ of translates of a centrally symmetric convex body $\mathbf{C}$ in $\mathbb{E}^d$ a $ρ$-separable packing for given $ρ\geq 1$ if in every ball concentric to a packing element of $\mathcal P$ having radius $ρ$ (measured in the norm generated by $\mathbf{C}$) the corresponding sub-packing of $\mathcal P$ is…
Optimal Extensions of Cross-Sections: Sphere Packings in Dimensions 38 to 43
State Key Lab of Processors, Institute of Computing Technology, Chinese Academy of Sciences（处理器重点实验室，中国科学院计算技术研究所）；University of Chinese Academy of Sciences（中国科学院大学）
Extending Blaschke and Lebesgue's classical result in the Euclidean plane, it has been recently proved in spherical and the hyperbolic cases, as well, that Reuleaux triangles have the minimal area among convex domains of constant width $D$. We prove an essentially optimal stability version of this statement in each of the three types of surfaces of constant curvature. In addition, we summarize the fundamental properties of convex bodies of constant width in spaces of constant curvature, and provide a characterization in the hyperbolic case in terms of horospheres.
It is well known that any measure in S^2 satisfying certain simple conditions is the surface measure of a bounded convex body in R^3. It is also known that a local perturbation of the surface measure may lead to a nonlocal perturbation of the corresponding convex body. We prove that, under mild conditions on a convex body, there are families of perturbations of its surface measure forming line segments, with the original measure at the midpoint, leading to local perturbations of the body. Moreover, there is, in a sense, a huge amount of such families. We apply this result to Newton's problem o…
In this paper we give an algorithm to round the floating point output of a semidefinite programming solver to a solution over the rationals or a quadratic extension of the rationals. We apply this to get sharp bounds for packing problems, and we use these sharp bounds to prove that certain optimal packing configurations are unique up to rotations. In particular, we show that the configuration coming from the $\mathsf{E}_8$ root lattice is the unique optimal code with minimal angular distance $π/3$ on the hemisphere in $\mathbb R^8$, and we prove that the three-point bound for the $(3, 8, \vart…
The intersection $L$ of two different non-opposite hemispheres of the unit sphere $S^2$ is called a lune. By $Δ(L)$ we denote the distance of the centers of the semicircles bounding $L$. By the thickness $Δ(C)$ of a convex body $C \subset S^2$ we mean the minimal value of $Δ(L)$ over all lunes $L \supset C$. We call a convex body $R\subset S^2$ reduced provided $Δ(Z) < Δ(R)$ for every convex body $Z$ being a proper subset of $R$. Our aim is to estimate the diameter of $R$, where $Δ(R) < \fracπ{2}$, in terms of its thickness.
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…
Massachusetts Institute of Technology（麻省理工学院）；UC Berkeley（加州大学伯克利分校）
We develop a technique for establishing lower bounds on the sample complexity of Least Squares (or, Empirical Risk Minimization) for large classes of functions. As an application, we settle an open problem regarding optimality of Least Squares in estimating a convex set from noisy support function measurements in dimension $d\geq 6$. Specifically, we establish that Least Squares is mimimax sub-optimal, and achieves a rate of $\tildeΘ_d(n^{-2/(d-1)})$ whereas the minimax rate is $Θ_d(n^{-4/(d+3)})$.
