On a quantitative reversal of Alexandrov's inequality

Alexandrov’s inequalities imply that for any convex body $A$, the sequence of intrinsic volumes $V_1(A),\dots ,V_n(A)$ is non-increasing (when suitably normalized). Milman’s random version of Dvoretzky’s theorem shows that a large initial segment of this sequence is essentially constant, up to a critical parameter called the Dvoretzky number. We show that this near-constant behavior actually extends further, up to a different parameter associated with $A$. This yields a new quantitative reverse inequality that sits between the approximate reverse Urysohn inequality, due to Figiel–Tomczak-Jaegermann and Pisier, and the sharp reverse Urysohn inequality for zonoids, due to Hug–Schneider. In fact, we study concentration properties of the volume radius and mean width of random projections of $A$ and show how these lead naturally to such reversals.