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

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

One of the maps may be chosen to be the identity.[3]

Notes
  1. Milman (1986)
  2. Pisier (1989)
  3. Pisier (1989)
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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]". Comptes Rendus de l'Académie des Sciences, Série I. 302 (1): 25–28. MR 0827101.CS1 maint: ref=harv (link)
  • 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.CS1 maint: ref=harv (link)
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