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We prove that there exists an absolute constant $alpha >1$ with the following property: if $K$ is a convex body in ${mathbb R}^n$ whose center of mass is at the origin, then a random subset $Xsubset K$ of cardinality ${rm card}(X)=lceilalpha nrceil $ satisfies with probability greater than $1-e^{-n}$ {Ksubseteq c_1n,{mathrm conv}(X),} where $c_1>0$ is an absolute constant. As an application we show that the vertex index of any convex body $K$ in ${mathbb R}^n$ is bounded by $c_2n^2$, where $c_2>0$ is an absolute constant, thus extending an estimate of Bezdek and Litvak for the symmetric case.
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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.
Let $K$ be an isotropic symmetric convex body in ${mathbb R}^n$. We show that a subspace $Fin G_{n,n-k}$ of codimension $k=gamma n$, where $gammain (1/sqrt{n},1)$, satisfies $$Kcap Fsubseteq frac{c}{gamma }sqrt{n}L_K (B_2^ncap F)$$ with probability greater than $1-exp (-sqrt{n})$. Using a different method we study the same question for the $L_q$-centroid bodies $Z_q(mu )$ of an isotropic log-concave probability measure $mu $ on ${mathbb R}^n$. For every $1leq qleq n$ and $gammain (0,1)$ we show that a random subspace $Fin G_{n,(1-gamma )n}$ satisfies $Z_q(mu )cap Fsubseteq c_2(gamma )sqrt{q},B_2^ncap F$. We also give bounds on the diameter of random projections of $Z_q(mu )$ and using them we deduce that if $K$ is an isotropic convex body in ${mathbb R}^n$ then for a random subspace $F$ of dimension $(log n)^4$ one has that all directions in $F$ are sub-Gaussian with constant $O(log^2n)$.
Central limit theorems for the log-volume of a class of random convex bodies in $mathbb{R}^n$ are obtained in the high-dimensional regime, that is, as $ntoinfty$. In particular, the case of random simplices pinned at the origin and simplices where all vertices are generated at random is investigated. The coordinates of the generating vectors are assumed to be independent and identically distributed with subexponential tails. In addition, asymptotic normality is established also for random convex bodies (including random simplices pinned at the origin) when the spanning vectors are distributed according to a radially symmetric probability measure on the $n$-dimensional $ell_p$-ball. In particular, this includes the cone and the uniform probability measure.
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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