Rigidity of Steiner's inequality for the anisotropic perimeter

TL;DR

This paper investigates the conditions under which the rigidity of Steiner's inequality for the anisotropic perimeter aligns with the Euclidean case, focusing on boundary normal vectors of the Steiner symmetral.

Contribution

It provides new conditions linking anisotropic and Euclidean rigidity, based on boundary normal vector restrictions, advancing understanding of extremals in anisotropic geometric inequalities.

Findings

Rigidity in anisotropic setting is equivalent to Euclidean rigidity under certain boundary conditions.

Conditions are expressed in terms of restrictions on boundary normal vectors.

Results unify anisotropic and Euclidean rigidity concepts for Steiner's inequality.

Abstract

The aim of this work is to study the rigidity problem for Steiner's inequality for the anisotropic perimeter, that is, the situation in which the only extremals of the inequality are vertical translations of the Steiner symmetral that we are considering. Our main contribution consists in giving conditions under which rigidity in the anisotropic setting is equivalent to rigidity in the Euclidean setting. Such conditions are given in term of a restriction to the possible values of the normal vectors to the boundary of the Steiner symmetral (see Corollary 1.17, and Corollary 1.18).