On the stability of Brunn–Minkowski type inequalities

Abstract

We establish the stability near a Euclidean ball of two conjectured inequalities: the dimensional Brunn-Minkowski inequality for radially symmetric log-concave measures in R n , and of the log-Brunn-Minkowski inequality.

Improved log-concavity for rotationally invariant measures of symmetric convex sets

“…This was improved to the optimal exponent 1 n in [9]. In [6] it was shown that (3) holds for arbitrary rotation invariant log-concave measures instead of γ, but only when K and L are small perturbations of a ball. In [18] Livshyts proved (3) for all even log-concave measures, but with the optimal exponent 1 n replaced with a worse exponent c n = 1 n 4+o(1) .…”

A universal bound in the dimensional Brunn-Minkowski inequality for log-concave measures

“…L, Marsiglietti, Nayar and Zvavitch [45] showed that this conjecture follows from the celebrated Log-Brunn-Minkowski conjecture of Böröczky, Lutwak, Yang, Zhang [8] (see also [9] and [10], and Milman [50], [51]); a combination of this result with the results of Saroglou [54], [55], confirms (4) for unconditional convex bodies and unconditional log-concave measures. For rotation-invariant measures, this conjecture was verified locally near any ball by Colesanti, L, Marsiglietti [16]. Kolesnikov, L [35] developed an approach to this question, building up on the past works of Kolesnikov and Milman [31], [32], [30], [34], as well as [16], and showed that in the case of the Gaussian measure, for convex sets containing the origin, the desired inequality holds with power 1/2n; this is curious, because earlier, Nayar and Tkocz [53] showed that only the assumption of the sets containing the origin is not sufficient for the inequality to hold with a power as strong as 1/n.…”

Mass Transportation Functionals on the Sphere with Applications to the Logarithmic Minkowski Problem

“…In addition, it was shown in [10] that the log-Brunn-Minkowski inequality has the following local version (see Proposition 4.4): u ∈ C 1 (S n−1 ) (see [31] for a more general version for p-Brunn-Minkowski inequality). (for n = 2).…”