Milman's reverse Brunn–Minkowski inequality
# Milman's reverse Brunn–Minkowski inequality

In mathematics, particularly, in asymptotic convex geometry, Milman's reverse Brunn–Minkowski inequality is a result due to Vitali Milman that provides a reverse inequality to the famous Brunn–Minkowski inequality for convex bodies in n-dimensional Euclidean space $R^n.$ Namely, it bounds the volume of the Minkowski sum of two bodies from above in terms of the volumes of the bodies.

## Introduction

Let K and L be convex bodies in $R^n.$ The Brunn–Minkowski inequality states that

    $\mathrm {vol} (K+L)^{1/n}\geq \mathrm {vol} (K)^{1/n}+\mathrm {vol} (L)^{1/n}~,$

where vol denotes n-dimensional Lebesgue measure and the + on the left-hand side denotes Minkowski addition.

In general, no reverse bound is possible, since one can find convex bodies K and L of unit volume so that the volume of their Minkowski sum is arbitrarily large. Milman's theorem states that one can replace one of the bodies by its image under a properly chosen volume-preserving linear map so that the left-hand side of the Brunn–Minkowski inequality is bounded by a constant multiple of the right-hand side.

The result is one of the main structural theorems in the local theory of Banach spaces.

## Statement of the inequality

There is a constant C, independent of n, such that for any two centrally symmetric convex bodies K and L in $R^n,$ there are volume-preserving linear maps φ and ψ from $R^n$ to itself such that for any real numbers s, t > 0

    $\mathrm {vol} (s\,\varphi K+t\,\psi L)^{1/n}\leq C\left(s\,\mathrm {vol} (\varphi K)^{1/n}+t\,\mathrm {vol} (\psi L)^{1/n}\right)~.$

One of the maps may be chosen to be the identity.

