Milman's reverse Brunn-Minkowski inequality

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In mathematics, 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 Rⁿ. At first sight, such a reverse inequality seems to be impossible, since if K and L are convex bodies with unit volume, the volume of their Minkowski sum K + L can be arbitrarily large. However, the use of volume-preserving linear maps allows one to prove Milman's reverse inequality, similarly to the reverse isoperimetric inequality. The result is also important in the local theory of Banach spaces.

[edit] 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

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

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

[edit] References

Gardner, Richard J. (2002). "The Brunn-Minkowski inequality". Bull. Amer. Math. Soc. (N.S.) 39 (3): pp. 355–405 (electronic). doi:10.1090/S0273-0979-02-00941-2. ISSN 0273-0979.
Milman, Vitali D. and Schechtman, Gideon (1986). Asymptotic theory of finite-dimensional normed spaces, Volume 1200 in Lecture Notes in Mathematics. Berlin: Springer-Verlag, pp. viii+156. ISBN 3-540-16769-2.