On the $L_p$-Brunn-Minkowski and dimensional Brunn-Minkowski conjectures for log-concave measures

Abstract

We study several of the recent conjectures in regards to the role of symmetry in the inequalities of Brunn-Minkowski type, such as the L p -Brunn-Minkowski conjecture of Böröczky, Lutwak, Yang and Zhang, and the Dimensional Brunn-Minkowski conjecture of Gardner and Zvavitch, in a unified framework. We obtain several new results for these conjectures.We show that when K ⊂ L, the multiplicative form of the L p -Brunn-Minkowski conjecture holds for Lebesgue measure for p ≥ 1−Cn −0.75 , which improves upon the estimate of Kolesnikov and Milman in the partial case when one body is contained in the other.We also show that the multiplicative version of the L p -Brunn-Minkowski conjecture for the standard Gaussian measure holds in the case of sets containing sufficiently large ball (whose radius depends on p). In particular, the Gaussian Log-Brunn-Minkowski conjecture holds when K and L contain 0.5(n + 1)B n 2 . We formulate an a-priori stronger conjecture for log-concave measures, extending both the L p -Brunn-Minkowski conjecture and the Dimensional one, and verify it in the case when the sets are dilates and the measure is Gaussian. We also show that the Log-Brunn-Minkowski conjecture, if verified, would yield this more general family of inequalities.Our results build up on the methods developed by Kolesnikov and Milman as well as Colesanti, Livshyts, Marsiglietti. We furthermore verify that the local version of these conjectures implies the global version in the setting of general measures, and this step uses methods developed recently by Putterman.

The Logarithmic Minkowski conjecture and the $L_p$-Minkowski Problem

“…We note that an analogues result holds for linear images of Hausdorff neighbourhoods of l q balls for q > 2 if the dimension n is high enough according to [158] and the method of [67]. Actually, Milman [191] provides rather generous explicit curvature pinching bounds for ∂K in order to Conjecture 3.2 to hold, and proves that for any origin symmetric convex body M there exists an origin symmetric convex body K with C ∞ + boundary and M ⊂ K ⊂ 8M such that Conjecture 3.2 holds for any origin symmetric convex body C. Additional local versions of Conjecture 3.2 are due to Colesanti, Livshyts, Marsiglietti [76], Kolesnikov, Livshyts [157] and Hosle, Kolesnikov, Livshyts [135]. We review Kolesnikov and Milman's approach in [158] based on the Hilbert-Brunn-Minkowski operator at the end of Section 4.…”

“…Concerning the L p Brunn-Minkowski conjecture, Hosle, Kolesnikov, Livshyts [135] and Kolesnikov, Livshyts [157] present certain natural generalizations and approaches.…”

Stable solution of the log-Minkowski problem in the case of many hyperplane symmetries

“…Another even more recent proof of this result based on Alexandrov's approach of considering the Hilbert-Brunn-Minkowski operator for polytopes is due to Putterman [60]. Additional local versions of Conjecture 3.1 are due to Kolesnikov, Livshyts [45] and Hosle, Kolesnikov, Livshyts [38].…”

Stability of the log-Brunn-Minkowski inequality in the case of many hyperplane symmetries

“…After initiating the study of the L p -Brunn-Minkowski inequality for a range of p by Firey [36] and Lutwak [55,56,57], major results have been obtained by Hug, Lutwak, Yang, Zhang [46], and more recently the papers Kolesnikov, Milman [52], Chen, Huang, Li, Liu [20], Hosle, Kolesnikov, Livshyts [45], Kolesnikov, Livshyts [51] present new developments and approaches.…”