The Alexandrov-Fenchel Inequality and its Extremal Structures

Abstract

The theory of convex bodies, developed by H. Minkowski (1903), began with the observation that the volume of a sum of convex bodies is a polynomial. The Alexandrov-Fenchel inequality (1937) established log-concavity relations between the coefficients of this polynomial. These relations have numerous applications extending beyond the theory of convex bodies itself. The characterization of the equality cases of this inequality has been unknown for more than 80 years. This thesis sheds new light on both the Alexandrov-Fenchel inequality itself and its equality cases. On the inequality side, we develop an analytic perspective on the Alexandrov-Fenchel inequality which leads to a new and simple proofs of it. On the equality side, we characterize the equality cases of the inequality for large classes of convex bodies, confirming conjectures of R. Schneider.