Volume inequalities for asymmetric Wulff shapes

TL;DR

This paper establishes new sharp reverse affine isoperimetric inequalities for asymmetric Wulff shapes and their polars, characterizing extremals and generalizing previous simplex inequalities, thus solving a problem posed by Zhang.

Contribution

The paper introduces new sharp reverse affine inequalities for asymmetric Wulff shapes, extending prior results and characterizing all extremal cases.

Findings

Established sharp reverse affine inequalities for asymmetric Wulff shapes

Characterized all extremal shapes achieving equality

Unified previous simplex inequalities as special cases

Abstract

Sharp reverse affine isoperimetric inequalities for asymmetric Wulff shapes and their polars are established, along with the characterization of all extremals. These new inequalities have as special cases previously obtained simplex inequalities by Ball, Barthe and Lutwak, Yang, and Zhang. In particular, they provide the solution to a problem by Zhang.