Georgia Institute of Technology College of Sciences

We will discuss certain isoperimetric-type problems for convex sets, such as the Log-Brunn-Minkowski conjecture for Lebesgue measure, and will explain the approach to this type of problems via local versions of inequalities and why it arises naturally. We consider a weaker form of the conjecture and prove it in several cases, with elementary geometric methods.  We shall also consider several illustrative ``hands on’’ examples. If time permits, we will discuss Bochner’s method approach to the question and formulate some new results in this regard. The second (optional!) part of this talk will be at the High-dimensional seminar right after, and will involve a discussion of more involved methods. Partially based on a joint work with Hosle and Kolesnikov.