Sharp inequalities for the mean distance of random points in convex bodies

For a convex body $K\subset {\mathbb{R}}^d$ the mean distance $\mathrm{\Delta }(K)=\mathbb{E}\mathrm{\mid }{X}_1-X_2\mathrm{\mid }$ is the expected Euclidean distance of two independent and uniformly distributed random points $X_1,X_2\in K$. Optimal lower and upper bounds for ratio between $\mathrm{\Delta }(K)$ and the first intrinsic volume $V_1(K)$ of $K$ (normalized mean width) are derived and degenerate extremal cases are discussed. The argument relies on Riesz’s rearrangement inequality and the solution of an optimization problem for powers of concave functions. The relation with results known from the existing literature is reviewed in detail.