ﻻ يوجد ملخص باللغة العربية
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.
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 measu
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 symmet
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: $$ f_{d,beta} (x) = text{const} cdot (1-|x|^2)^{beta}, quad |x|leq 1, quad text{(the beta case)} $$ an
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 $d=3$,
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