Towards an entropy-based sumset calculus for additive combinatorics and convex geometry
Seminar Room 2, Newton Institute Gatehouse
We use common entropy-based tools to study two kinds of problems: the first of proving general cardinality inequalities for sumsets in possibly nonabelian groups, and the second of proving volume inequalities of interest in convex geometry and geometric functional analysis. We will spend most of our time on the discrete setting (joint work with A. Marcus and P. Tetali), introducing the notion of partition-determined functions, and presenting some basic new inequalities for the entropy of such functions of independent random variables, as well as for cardinalities of compound sets obtained using these functions. Corollaries of the results for partition-determined functions include entropic analogues of general Pl\"unnecke-Ruzsa type inequalities, sumset cardinality inequalities in abelian groups generalizing inequalities of Gyarmati-Matolcsi-Ruzsa and Balister-Bollob\'as, and partial progress towards a conjecture of Ruzsa for sumsets in nonabelian groups. Time permitting, we will also mention some results in the continuous setting (joint work with S. Bobkov) including some Pl\"unnecke-type inequalities for Minkowski sums of convex sets, and an entropic generalization of V. Milman's reverse Brunn-Minkowski inequality.