On the Constant in the Reverse Brunn-Minkowski Inequality for p-Convex Balls

Summary

This note is devoted to the study of the dependence on p of the constant in the reverse Brunn-Minkowski inequality for p-convex balls (that is, p-convex symmetric bodies). We will show that this constant is estimated as for absolute constants c > 1 and C>1.

Recall that a p-norm on real vector space X is a map 11·11 : X -+ ℝ⁺ satisfying these conditions:

• (1) ||xll > 0 for all x ≠ 0.

• (2) ||tx|| = |t|||x|| for all t ∈ ℝ and x ∈ X.

• (3) ||x + y||^P ≤ ||x||^P + ||y||^P for all x, y ∈X.

Note that the unit ball of p-normed space is a p-ball and, vice versa, the gauge of p-ball is a p-norm.

Recently, J. Bastero, J. Bernues, and A. Peiia [BBP] extended the reverse Brunn-Minkowski inequality, which was discovered by V. Milman [MJ, to the class of p-convex balls. They proved the following result.