  For a convex body $K\subset\mathbb{R}^d$ the mean distance
$\Delta(K)=\mathbb{E}|X_1-X_2|$ 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 $\Delta(K)$ 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.
