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Register a new userApproximation of smooth convex bodies by random polytopes
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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.
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The deviation of a general convex body with twice differentiable boundary and an arbitrarily positioned polytope with a given number of vertices is studied. The paper considers the case where the deviation is measured in terms of the surface areas of the involved sets, more precisely, by what is called the surface area deviation. The proof uses arguments and constructions from probability, convex and integral geometry. The bound is closely related to $p$-affine surface areas.
Let $K subset R^d$ be a smooth convex set and let $P_la$ be a Poisson point process on $R^d$ of intensity $la$. The convex hull of $P_la cap K$ is a random convex polytope $K_la$. As $la to infty$, we show that the variance of the number of $k$-dimensional faces of $K_la$, when properly scaled, converges to a scalar multiple of the affine surface area of $K$. Similar asymptotics hold for the variance of the number of $k$-dimensional faces for the convex hull of a binomial process in $K$.
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.
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.
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 $O(log^2n)$.
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