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We study the expected volume of random polytopes generated by taking the convex hull of independent identically distributed points from a given distribution. We show that for log-concave distributions supported on convex bodies, we need at least exponentially many (in dimension) samples for the expected volume to be significant and that super-exponentially many samples suffice for concave measures when their parameter of concavity is positive.
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Let $X_1,ldots,X_N$, $N>n$, be independent random points in $mathbb{R}^n$, distributed according to the so-called beta or beta-prime distribution, respectively. We establish threshold phenomena for the volume, intrinsic volumes, or more general measures of the convex hulls of these random point sets, as the space dimension $n$ tends to infinity. The dual setting of polytopes generated by random halfspaces is also investigated.
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 $X_1,ldots,X_n$ be i.i.d. random points in the $d$-dimensional Euclidean space sampled according to one of the following probability densities: $$ f_{d,beta} (x) = text{const} cdot (1-|x|^2)^{beta}, quad |x|leq 1, quad text{(the beta case)} $$ and $$ tilde f_{d,beta} (x) = text{const} cdot (1+|x|^2)^{-beta}, quad xinmathbb{R}^d, quad text{(the beta case).} $$ We compute exactly the expected intrinsic volumes and the expected number of facets of the convex hull of $X_1,ldots,X_n$. Asymptotic formulae where obtained previously by Affentranger [The convex hull of random points with spherically symmetric distributions, 1991]. By studying the limits of the beta case when $betadownarrow -1$, respectively $beta uparrow +infty$, we can also cover the models in which $X_1,ldots,X_n$ are uniformly distributed on the unit sphere or normally distributed, respectively. We obtain similar results for the random polytopes defined as the convex hulls of $pm X_1,ldots,pm X_n$ and $0,X_1,ldots,X_n$. One of the main tools used in the proofs is the Blaschke-Petkantschin formula.
Consider a random set of points on the unit sphere in $mathbb{R}^d$, which can be either uniformly sampled or a Poisson point process. Its convex hull is a random inscribed polytope, whose boundary approximates the sphere. We focus on the case $d=3$, for which there are elementary proofs and fascinating formulas for metric properties. In particular, we study the fraction of acute facets, the expected intrinsic volumes, the total edge length, and the distance to a fixed point. Finally we generalize the results to the ellipsoid with homeoid density.
A question related to some conjectures of Lutwak about the affine quermassintegrals of a convex body $K$ in ${mathbb R}^n$ asks whether for every convex body $K$ in ${mathbb R}^n$ and all $1leqslant kleqslant n$ $$Phi_{[k]}(K):={rm vol}_n(K)^{-frac{1}{n}}left (int_{G_{n,k}}{rm vol}_k(P_F(K))^{-n},d u_{n,k}(F)right )^{-frac{1}{kn}}leqslant csqrt{n/k},$$ where $c>0$ is an absolute constant. We provide an affirmative answer for some broad classes of random polytopes. We also discuss upper bounds for $Phi_{[k]}(K)$ when $K=B_1^n$, the unit ball of $ell_1^n$, and explain how this special instance has implications for the case of a general unconditional convex body $K$.
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