Remarks on the conjectured log-Brunn–Minkowski inequality

Abstract

Böröczky, Lutwak, Yang and Zhang recently conjectured a certain strengthening of the Brunn-Minkowski inequality for symmetric convex bodies, the so-called log-Brunn-Minkowski inequality. We establish this inequality together with its equality cases for pairs of convex bodies that are both unconditional with respect to some orthonormal basis. Applications of this fact are discussed. Moreover, we prove that the log-Brunn-Minkowski inequality is equivalent to the (B)-Theorem for the uniform measure of the cube (this has been proven by Cordero-Erasquin, Fradelizi and Maurey for the gaussian measure instead).

The log-Brunn-Minkowski inequality in ℝ³

“…Remark 3.4. Theorems 3.2 and 3.3 implies that inequalities (3.3) and (3.5) hold when K is in R 1 class with respect to L, and Saroglou [24] proved that inequalities (3.3) and (3.5) hold when K and L are unconditional. Next, we show that these two conditions don't include each other.…”

“…Ma [20] gave an alternative proof of (1.1) for the case n = 2. Saroglou [24] established (1.2) together with its equality cases for pairs of convex bodies that are both unconditional with respect to some orthonormal basis. Stancu [28] showed some variants of the logarithmic Minkowski inequality for general convex bodies and obtained some special cases for the equality holds in (1.1) without the symmetric assumption.…”

The log-Brunn–Minkowski inequality

“…The Log-Brunn-Minkowski Conjecture 1 and Log-Minkowski Conjecture 2 were verified for unconditional convex bodies (in a slightly stronger form for coordinatewise products, see the Appendix Section 7) by several authors like Bollobas, Leader [6] and Cordero-Erausquin, Fradelizi, Maurey [29] even before the log-Brunn-Minkowski conjecture was stated, and the equality case was described by Saroglou [83]. Actually, the paper [83] contains a small gap concerning the equality case, and we clarify the argument in the Appendix Section 7.…”

Copolar convexity

“…We would like to mention here a few variants of the inequality. In the first one (proved under certain symmetry conditions), the Minkowski combinations P t are replaced by certain log-Minkowski combinations [3], [20], [21]. Another version considers, instead of the combinations P t , balls in the norms of the complex Calderón interpolated spaces [4].…”