Optimal mean width and metric entropy estimates for convex bodies

We show that there exists a constant $C > 0$ such that for any $n \geq 1$ and any convex body $K \subset \mathbf{R}^n$, \[ 1 \leq \inf_{T \in \mathrm{GL}(n)} \, \frac{M^\ast(TK)}{\mathrm{vr}(TK)} \leq C\sqrt{\log(\mathrm{e} n)}, \] where $M^\ast$ denotes the spherical mean width and $\mathrm{vr}(\cdot)$ denotes the volume radius. The righthand side is attained, up to universal constants, by the crosspolytope and the regular $n$-simplex. Analogously, we show that, up to universal constants, the logarithm of the Euclidean covering number is maximized over convex bodies $K \subset \mathbf{R}^n$ by the simplex and crosspolytope. Our proof makes use of Eldan's stochastic localization.