ar X iv : 1 11 0 . 65 84 v 2 [ m at h . M G ] 6 A ug 2 01 2 Dimensionality and the stability of the Brunn - Minkowski inequality
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چکیده
for any compact sets K, T ⊂ R, where (K +T )/2 = {(x+ y)/2; x ∈ K, y ∈ T} is half of the Minkowski sum of K and T , and where V oln stands for the Lebesgue measure in R. Equality in (1) holds if and only if K is a translate of T and both are convex, up to a set of measure zero. The literature contains various stability estimates for the Brunn-Minkowski inequality, which imply that when there is almost-equality in (1), then K and T are almost-translates of each other. Such estimates appear in Diskant [8], in Groemer [13], and in Figalli, Maggi and Pratelli [11, 12]. We recommend Osserman [20] for a general survey on the stability of geometric inequalities. All of the stability results that we found in the literature share a common feature: Their estimates deteriorate quickly as the dimension increases. For instance, suppose that K, T ⊂ R are convex sets with
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تاریخ انتشار 2012