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
- [math]\displaystyle{ \mathrm{vol}(K+L)^{1/n} \geq \mathrm{vol}(K)^{1/n} + \mathrm{vol}(L)^{1/n}~, }[/math]
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
- [math]\displaystyle{ \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)~. }[/math]
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]". Comptes Rendus de l'Académie des Sciences, Série I 302 (1): 25–28.
- 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.
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