Asymptotic mean values of Gaussian polytopes
As a consequence of more general results the authors show the following interesting theorem: Let \(X_1,\dots , X_n\) be i.i.d.\ random points in \(\mathbb R^d\) with the common Gaussian normal distribution. Then for the expectation of the \(k\)-dimensional measure of the \(k\)-skeleton of the random polytope \(\text{conv}\{ X_1,\dots ,X_n\}\) and for the expectation of the number of \(k\)-dimensional faces of this polytope the following asymptotic formulae hold: \(E\, V_k (\text{skel}_k\, \text{conv}\{ X_1,\dots , X_n\})\sim c_{k,d}(\log n)^{d-1\over 2},\) \(E\, f_k\, \text{conv}\{X_1,\dots , X_n\})\sim e_{k,d}(\log n)^{d-1\over 2}\) as \(n\to\infty\) with suitable constants \(c_{k,d}\) and \(e_{kd}\) depending only on \(k\) and \(d\). Explicit representations for these constants are given. The second result is related to a result of \textit{F. Affentranger} and \textit{R. Schneider} [Discrete Comput.\ Geom.~7, 219--226 (1992; Zbl 0751.52002)] on random polytopes in the sense of Goodman and Pollack, as follows from a result of \textit{Y. M. Baryshnikov} and \textit{R. A. Vitale} [Discrete Comput.\ Geom.~11, 141--147 (1994; Zbl 0795.52002)].
scientific article; zbMATH DE number 2100241