Remarks on an inequality of Rogers and Shephard

Abstract:

A classical inequality of Rogers and Shephard states that if $K$ is a centered convex body of volume $1$ in ${\mathbb R}^n$, then \begin{equation*}1\leqslant g(K,k;F):=\big (\operatorname {vol}_k(P_F(K)) \operatorname {vol}_{n-k}(K\cap F^{\perp })\big )^{1/k} \leqslant {n\choose k}^{1/k}\leqslant \frac {cn}{k}\end{equation*} for every $F\in G_{n,k}$, where $c>0$ is an absolute constant. We show that if $K$ is origin symmetric and isotropic, then, for every $1\leqslant k\leqslant n-1$, a random $F\in G_{n,k}$ satisfies \begin{equation*}c_1L_K^{-1}\sqrt {n/k}\leqslant g(K,k;F)\leqslant c_2\sqrt {n/k}\ (\log n)^2 L_K\end{equation*} with probability greater than $1-e^{-k}$, where $L_K$ is the isotropic constant of $K$ and $c_1,c_2>0$ are absolute constants.