19 jun 2014 -- 15:10 [open in google calendar]
We discuss elementary proofs of sharp isoperimetric inequalities on a normed space equipped with a homogeneous measure. When the degree of homogeneity is positive, we use the Borell-Brascamp-Lieb inequality to present a new proof of a result by Cabre, Ros-Oton and Serra. However, it turns out that when the degree of homogeneity is negative, the relevant property is a new ”Complemented Concavity” property. We will define this new notion, and explain why every homogeneous measure satisfies it. This will allow us to present a new isoperimetric inequality in the ”negative” case, extending results by Canete and Rosales and by Howe. We will conclude by discussing the case of non-homogeneous measures, and explain how the homogeneity may be replaced by a suitable log-convexity assumption. Based on joint work with Emanuel Milman.