On the analogue of the concavity of entropy power in the Brunn-Minkowski theory (Q403156)

As a counterpart of the entropy power with added Gaussian noise in information theory, \textit{M. H. M. Costa} and \textit{T. M. Cover} posed in [IEEE Trans. Inf. Theory 30, 837--839 (1984; Zbl 0557.94006)] the following conjecture in the context of the Brunn-Minkowski theory.NEWLINENEWLINEConjecture: Let \(A\) be a bounded measurable set in \(\mathbb{R}^n\), then the function \(t\mapsto |A+tB^n_2|^{1/n}\) is concave in \(\mathbb{R}_+\). Here \(|\cdot|\) denotes the volume and \(B_n^2\) is the \(n\)-dimensional unit ball.NEWLINENEWLINEThe main result in this work provides an answer to the above conjecture depending on the dimension. The authors approach also the more general case in which the unit ball is replaced by a (fixed) compact and convex set, proving in this setting that the conjecture is true in dimension \(1\). In dimension \(2\) they prove that the conjecture is true for connected sets, and that for \(n\geq 2\) there exist non connected sets for which the conjecture fails. In dimension \(n\geq 3\), it is proven that connectivity is not enough being the conjecture false in general.NEWLINENEWLINEThe authors establish equivalent formulations of the Costa and Cover conjecture, linking it directly to the Brunn-Minkowski inequality. They also study its relation with other geometric inequalities.