  Alexandrov's inequalities imply that for any convex body $A$, the sequence of
intrinsic volumes $V_1(A),\ldots,V_n(A)$ 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.
