A Shannon-Kneser-Poulsen theorem

Consider the following questions.
Question 1: Does the volume of a union of balls decrease when their centres are brought pairwise closer?
Question 2: Does communication over an additive white Gaussian noise channel worsen when the transmitters are brought pairwise closer?
These questions appeal to our basic intuition about geometry and information transmission, which seems to suggest the answer to both of them is yes. The first question is open; the Kneser-Poulsen conjecture asserts that it has an affirmative answer. In this talk, based on well-known analogies between convex geometry and information theory, we will frame (and prove) the natural entropic analogue of the Kneser-Poulsen conjecture. As a corollary, an affirmative answer to the second question is obtained. Moreover, we will see how an elementary combination of our result with Costa’s entropy power inequality yields a unified strengthening of both statements. This talk is based on joint work with Dongbin Li.