Monotonicity of functionals of random polytopes

The convex hull $P_n$ of a Gaussian sample $X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n$ in $R^d$ is a Gaussian polytope. We prove that the expected number of facets $E f_{d-1} (P_n)$ismonotonicallyincreasingin$n$. Furthermore we prove this for random polytopes generated by uniformly distributed points in a $d$-dimensional ball.