Nicola Fusco

Università di Napoli Federico II

The isoperimetric inequality outside a convex set: the case of equality

In 2007 Choe, Ghomi and Ritoré proved the relative isoperimetric inequality outside a convex set \(C\). Precisely, they showed that if \(E\subset{\mathbb R}^N\setminus C\) has finite mass \(m\), then the perimeter of \(E\) outside \(C\) is greater than or equal to one half the surface measure of the ball with mass \(2m\). Moreover, they showed that if \(C\) is smooth and the equality holds in this isoperimetric inequality, then \(E\) is a half ball sitting on \(C\). In this talk I will discuss the equality case when \(C\) is any convex set, not necessarily smooth. This is a joint work with M. Morini.