Classification of equality cases in the classical Alexandrov–Fenchel inequality

Characterize all tuples of convex bodies (K1, K2, K3, …, K_{n+1}) in R^{n+1} for which equality holds in the classical Alexandrov–Fenchel inequality V^2(K1, K2, K3, …, K_{n+1}) = V(K1, K1, K3, …, K_{n+1}) · V(K2, K2, K3, …, K_{n+1}), thereby determining the complete set of extremals for the Alexandrov–Fenchel inequality for mixed volumes.

Background

The paper reviews the classical Alexandrov–Fenchel inequality for mixed volumes V(K1, K2, …, K_{n+1}) and discusses known representations via support functions when the bodies have C^2 boundaries. While the inequality itself is classical, understanding exactly when equality occurs (i.e., classifying extremals) has been a long-standing challenge in convex geometry.

The authors note recent progress by Shenfeld and van Handel, who classified the extremals in important special cases, including convex polytopes and certain combinations of polytopes, zonoids, and smooth bodies. However, the general classification problem remains open, and the specific capillary setting treated in this paper is not directly covered by those results. The present work establishes a capillary version of the Alexandrov–Fenchel inequality and its rigidity, but does not resolve the full general classification in the classical setting.