# On a quantitative reversal of Alexandrov's inequality

## TL;DR

This paper extends the understanding of intrinsic volumes of convex bodies, establishing a new quantitative reverse inequality that broadens the known near-constant behavior of these volumes beyond the Dvoretzky number.

## Contribution

It introduces a novel reverse inequality for intrinsic volumes, extending the range of near-constant behavior and connecting it with existing inequalities for convex bodies.

## Key findings

- Intrinsic volumes exhibit extended near-constant behavior.
- New reverse inequality sits between known inequalities for convex bodies.
- Concentration properties of volume radius and mean width lead to reversals.

## Abstract

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.