### A nonabelian Brunn-Minkowski inequality

Linfoot Number Theory Seminar

8th March 2023, 11:00 am – 12:00 pm

Fry Building, Room 2.04

The celebrated Brunn-Minkowski inequality states that for compact subsets X and Y of R^d, m(X+Y)^1/d ≥ m(X)^1/d+m(Y)^1/d where m(⋅) is the Lebesgue measure. We will introduce a conjecture generalizing this inequality to every locally compact group where the exponent is believed to be sharp. In a joint work with Chieu-Minh Tran and Ruixiang Zhang, we prove this conjecture for a large class of groups (including e.g. all real linear algebraic groups, and solvable groups). We also prove that the general conjecture will follow from the simple Lie group case. For those groups where we do not know the conjecture yet (one typical example being the universal covering of SL2(R)), we also obtain partial results, which are of the correct order of magnitude. In this talk, I will discuss this inequality and explain important ingredients, old and new, in our proof.

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