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Register a new userApproximations of convex bodies by measure-generated sets
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Given a Borel measure $mu$ on ${mathbb R}^{n}$, we define a convex set by [ M({mu})=bigcup_{substack{0le fle1, int_{{mathbb R}^{n}}f,{rm d}{mu}=1 } }left{ int_{{mathbb R}^{n}}yfleft(yright),{rm d}{mu}left(yright)right} , ] where the union is taken over all $mu$-measurable functions $f:{mathbb R}^{n}toleft[0,1right]$ with $int_{{mathbb R}^{n}}f,{rm d}{mu}=1$. We study the properties of these measure-generated sets, and use them to investigate natural variations of problems of approximation of general convex bodies by polytopes with as few vertices as possible. In particular, we study an extension of the vertex index which was introduced by Bezdek and Litvak. As an application, we provide a lower bound for certain average norms of centroid bodies of non-degenerate probability measures.
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The $K$-hull of a compact set $Asubsetmathbb{R}^d$, where $Ksubset mathbb{R}^d$ is a fixed compact convex body, is the intersection of all translates of $K$ that contain $A$. A set is called $K$-strongly convex if it coincides with its $K$-hull. We propose a general approach to the analysis of facial structure of $K$-strongly convex sets, similar to the well developed theory for polytopes, by introducing the notion of $k$-dimensional faces, for all $k=0,dots,d-1$. We then apply our theory in the case when $A=Xi_n$ is a sample of $n$ points picked uniformly at random from $K$. We show that in this case the set of $xinmathbb{R}^d$ such that $x+K$ contains the sample $Xi_n$, upon multiplying by $n$, converges in distribution to the zero cell of a certain Poisson hyperplane tessellation. From this results we deduce convergence in distribution of the corresponding $f$-vector of the $K$-hull of $Xi_n$ to a certain limiting random vector, without any normalisation, and also the convergence of all moments of the $f$-vector.
Let $K$ be a convex body in $mathbb{R}^n$ and $f : partial K rightarrow mathbb{R}_+$ a continuous, strictly positive function with $intlimits_{partial K} f(x) d mu_{partial K}(x) = 1$. We give an upper bound for the approximation of $K$ in the symmetric difference metric by an arbitrarily positioned polytope $P_f$ in $mathbb{R}^n$ having a fixed number of vertices. This generalizes a result by Ludwig, Schutt and Werner $[36]$. The polytope $P_f$ is obtained by a random construction via a probability measure with density $f$. In our result, the dependence on the number of vertices is optimal. With the optimal density $f$, the dependence on $K$ in our result is also optimal.
The paper is devoted to coverings by translative homothets and illuminations of convex bodies. For a given positive number $alpha$ and a convex body $B$, $g_{alpha}(B)$ is the infimum of $alpha$-powers of finitely many homothety coefficients less than 1 such that there is a covering of $B$ by translative homothets with these coefficients. $h_{alpha}(B)$ 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 $alpha$. In this paper, we prove that $g_{alpha}(B)leq h_{alpha}(B)$, find upper and lower bounds for both numbers, and discuss several general conjectures. In particular, we show that $h_{alpha} (B) > 2^{d-alpha}$ for almost all $alpha$ and $d$ when $B$ is the $d$-dimensional cube, thus disproving the conjecture from Research Problems in Discrete Geometry by Brass, Moser, and Pach.
We study a long standing open problem by Ulam, which is whether the Euclidean ball is the unique body of uniform density which will float in equilibrium in any direction. We answer this problem in the class of origin symmetric n-dimensional convex bodies whose relative density to water is 1/2. For n=3, this result is due to Falconer.
Barker and Larman asked the following. Let $K subset {Bbb{R}}^d$ be a convex body, whose interior contains a given convex body $K subset {Bbb{R}}^d$, and let, for all supporting hyperplanes $H$ of $K$, the $(d-1)$-volumes of the intersections $K cap H$ be given. Is $K$ then uniquely determined? Yaskin and Zhang asked the analogous question when, for all supporting hyperplanes $H$ of $K$, the $d$-volumes of the caps cut off from $K$ by $H$ are given. We give local positive answers to both of these questions, for small $C^2$-perturbations of $K$, provided the boundary of $K$ is $C^2_+$. In both cases, $(d-1)$-volumes or $d$-volumes can be replaced by $k$-dimensional quermassintegrals for $1 le k le d-1$ or for $1 le k le d$, respectively. Moreover, in the first case we can admit, rather than hyperplane sections, sections by $l$-dimensional affine planes, where $1 le k le l le d-1$. In fact, here not all $l$-dimensional affine subspaces are needed, but only a small subset of them (actually, a $(d-1)$-manifold), for unique local determination of $K$.
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