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Register a new userOn the monotonicity of the moments of volumes of random simplices
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In a $d$-dimensional convex body $K$ random points $X_0, dots, X_d$ are chosen. Their convex hull is a random simplex. The expected volume of a random simplex is monotone under set inclusion, if $K subset L$ implies that the expected volume of a random simplex in $K$ is smaller than the expected volume of a random simplex in $L$. Continuing work of Rademacher, it is shown that moments of the volume of random simplices are in general not monotone under set inclusion.
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In a $d$-dimensional convex body $K$, for $n leq d+1$, random points $X_0, dots, X_{n-1}$ are chosen according to the uniform distribution in $K$. Their convex hull is a random $(n-1)$-simplex with probability $1$. We denote its $(n-1)$-dimensional volume by $V_{K[n]}$. The $k$-th moment of the $(n-1)$-dimensional volume of a random $(n-1)$-simplex is monotone under set inclusion, if $K subseteq L$ implies that the $k$-th moment of $V_{K[n]}$ is not larger than that of $V_{L[n]}$. Extending work of Rademacher [On the monotonicity of the expected volume of a random simplex. Mathematika 58 (2012), 77--91] and Reichenwallner and Reitzner [On the monotonicity of the moments of volumes of random simplices. Mathematika 62 (2016), 949--958], it is shown that for $n leq d$, the moments of $V_{K[n]}$ are not monotone under set inclusion.
The sample range of uniform random points $X_1, dots , X_n$ chosen in a given convex set is the convex hull ${rm conv}[X_1, dots, X_n]$. It is shown that in dimension three the expected volume of the sample range is not monotone with respect to set inclusion. This answers a question by Meckes in the negative. The given counterexample is the three-dimensional tetrahedron together with an infinitesimal variation of it. As side result we obtain an explicit formula for all even moments of the volume of a random simplex which is the convex hull of three uniform random points in the tetrahedron and the center of one facet.
Let $X_1,ldots,X_n$ be independent random points that are distributed according to a probability measure on $mathbb{R}^d$ and let $P_n$ be the random convex hull generated by $X_1,ldots,X_n$ ($ngeq d+1$). Natural classes of probability distributions are characterized for which, by means of Blaschke-Petkantschin formulae from integral geometry, one can show that the mean facet number of $P_n$ is strictly monotonically increasing in $n$.
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.
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.
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