On the analogue of the concavity of entropy power in the Brunn–Minkowski theory

Abstract

Elaborating on the similarity between the entropy power inequality and the Brunn-Minkowski inequality, Costa and Cover conjectured in On the similarity of the entropy power inequality and the BrunnMinkowski inequality (IEEE Trans. Inform. Theory 30 (1984), no. 6, 837-839) the 1 n -concavity of the outer parallel volume of measurable sets as an analogue of the concavity of entropy power. We investigate this conjecture and study its relationship with geometric inequalities.

On the improvement of concavity of convex measures

“…One can then construct an explicit counterexample in every dimension n ≥ 2. Moreover, the counterexample in [18] shows more than inequality (6). It shows that…”

“…In the case of non-convex sets, this concavity property is false in general, even for the classical parallel volume |A + tB n 2 |. However, some conditions are given on A in [6] for which the parallel volume of A is 1 n -concave on R + . Notice that other concavity properties of generalized forms of the classical parallel volume have been established in [17].…”

On the Brunn-Minkowski inequality for general measures with applications to new isoperimetric-type inequalities

“…The authors conjectured that for any measurable set A the parallel volume is 1/n-concave. In [10], M. Fradelizi and the second named author proved that this conjecture is true for any measurable set in dimension 1 and for any connected set in dimension 2. However, the authors proved that this conjecture fails for arbitrary sets in dimension n ≥ 2.…”

Forward and Reverse Entropy Power Inequalities in Convex Geometry

“…However, this turns out not to be the case. It was shown by Fradelizi and Marsiglietti [68] that the analogue of Costa's result (19) on concavity of entropy power, namely the assertion that t → |A + tB d 2 | 1 d is concave for positive t and any given Borel set A, fails to hold 5 even in dimension 2. Another conjecture in this spirit that was made independently by V. Milman (as a generalization of Bergstrom's determinant inequality) and by Dembo, Cover and Thomas [56] (as an analogue of Stam's Fisher information inequality, which is closely related to the EPI) was disproved by Fradelizi, Giannopoulos and Meyer [63].…”