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

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