Minkowski Inequalities via Nonlinear Potential Theory

Florian Babisch 

In 1903, Hermann Minkowski, in his work 'Volumen und Oberflächen,' established two inequalities for convex bodies. The first inequality asserts that among all convex bodies with the same surface area, balls maximize the product of volume and the integral of mean curvature. The second inequality, now known as the Minkowski Inequality, states that balls alone minimize the integral of mean curvature. Consequently, these inequalities imply that balls have the largest volume among all convex bodies with the same surface area.

In this talk, I will provide a brief historical overview and recall relevant concepts from differential geometry. Following this, I will introduce the $L^p$-Minkowski Inequality and explain its significance. Subsequently, I will provide a concise overview of techniques previously employed to prove this inequality, and I will elucidate the approach used by Agostiniani, Fogagnolo, and Mazzieri, along with essential concepts necessary for understanding it.