On determinants and the volume of random polytopes in isotropic convex bodies (Q605058)

The author considers random polytopes generated by sampling vectors in multiple convex bodies. Let \(K^{(1)},\dots, K^{(n)}\) be isotropic convex bodies in \(\mathbb{R}^n\), with isotropic constants \(L_{ K^{(i)}}\), and let \(X_1,\dots,X_n\) be independent random vectors such that \(X_i\) is uniformly distributed in \(K^{(i)}\), for \(i=1,\dots,n\). A lower bound containing explicitly the constants \(L_{K^{(i)}}\) is determined for the volume of the random polytope formed by the absolute convex hull \(K_n:=\mathrm{conv}\{\pm X_1,\dots,\pm X_n\}\) of the selected vectors. The result is proved from the log-concave measure point of view, and several applications are discussed.