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^[1] that provides a reverse inequality to the famous Brunn–Minkowski inequality for convex bodies in n-dimensional Euclidean space Rⁿ. 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 Rⁿ. 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.^[2]

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 Rⁿ, there are volume-preserving linear maps φ and ψ from Rⁿ 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.^[3]

Notes

References

Milman, Vitali D. (1986). "Inégalité de Brunn-Minkowski inverse et applications à la théorie locale des espaces normés. [An inverse form of the Brunn-Minkowski inequality, with applications to the local theory of normed spaces]". C. R. Acad. Sci. Paris Sér. I Math. 302 (1): 25–28. MR 0827101.

Pisier, Gilles (1989). The volume of convex bodies and Banach space geometry. Cambridge Tracts in Mathematics 94. Cambridge: Cambridge University Press. ISBN 0-521-36465-5. MR 1036275.