Random spaces generated by vertices of the cube (Q1864192)

Let \(E^n_2= \{-1,1\}^n\) be the vertex set of the cube \([-1,1]\) in \(\mathbb{R}^n\). For \(N\geq n\), the authors consider the absolute convex hulls \(K_n= \text{conv}\{\pm x_1,\dots,\pm x_N\}\) of \(N\) random points \(x_1,\dots, x_N\), chosen independently and uniformly from \(E^n_2\). They show various asymptotic properties of the Banach spaces \(X_N\) generated by the random polytopes \(K_N\), which hold with high probability, as \(n\), \(N\to\infty\). The results are based on inequalities for geometric functionals which are shown first. The functionals considered include the inradius, the volume radius, the mean width and the size of the maximal inscribed cube.