Open Collections

Description

Given a convex body $K$ and a real number $a$, how should we define the convex body $K^a$? One way to answer this question is to write down a list of natural properties we expect from such a power, and then prove existence and uniqueness of a construction satisfying these properties. In this talk we will explain why a natural power operation does not exist for $a > 1$, but does exist for $0 < a < 1$. We will also discuss the uniqueness question, which is more delicate. Based on joint work with Vitali Milman.