Anisotropic extrinsic radius pinching for hypersurfaces and the stability of the Wulff shape

TL;DR

This paper establishes an inequality relating anisotropic extrinsic radius and mean curvatures of hypersurfaces, showing that near equality implies the hypersurface is close to the Wulff shape, thus contributing to geometric stability analysis.

Contribution

It extends the Hasanis--Koutroufiotis inequality to anisotropic settings and analyzes the stability of the Wulff shape under near-extremal conditions.

Findings

Proved the anisotropic inequality for hypersurfaces.

Characterized the equality case and stability near the Wulff shape.

Showed that almost extremal hypersurfaces are close to the Wulff shape in Hausdorff distance.

Abstract

We prove the Hasanis--Koutroufiotis type inequality for the anisotropic extrinsic radius of hypersurfaces in Euclidean space involving the anisotropic mean curvatures. We also study the equality case and proved that an almost extremal hypersurface must be close to the Wulff shape in the sense of the Hausdorff distance.