Variance asymptotics for random polytopes in smooth convex bodies

Let $K subset mathbb R ^d$ be a smooth convex set and let $mathcal{P }_{lambda }$ be a Poisson point process on $mathbb R ^d$ of intensity ${lambda }$ . The convex hull of $mathcal{P }_{lambda }cap K$ is a random convex polytope $K_{lambda }$ . As ${lambda }rightarrow infty $ , we show that the variance of the number of $k$ -dimensional faces of $K_{lambda }$ , when properly scaled, converges to a scalar multiple of the affine surface area of $K$ . Similar asymptotics hold for the variance of the number of $k$ -dimensional faces for the convex hull of a binomial process in $K$ .