Extremizers and stability of the Betke--Weil inequality

TL;DR

This paper proves that the only convex domain achieving equality in a specific geometric inequality involving mixed area and perimeter is the regular triangle, confirming a conjecture and establishing a stability result.

Contribution

It confirms that the regular triangle uniquely attains equality in the Betke--Weil inequality among all convex domains, strengthening the understanding of extremizers and stability.

Findings

Equality holds only for the regular triangle.

Established a stronger stability result for the inequality.

Confirmed the conjecture by Betke and Weil.

Abstract

Let $K$ be a compact convex domain in the Euclidean plane. The mixed area $A(K,-K)$ of $K$ and $-K$ can be bounded from above by $1/(6\sqrt{3})L(K)^2$, where $L(K)$ is the perimeter of $K$. This was proved by Ulrich Betke and Wolfgang Weil (1991). They also showed that if $K$ is a polygon, then equality holds if and only if $K$ is a regular triangle. We prove that among all convex domains, equality holds only in this case, as conjectured by Betke and Weil. This is achieved by establishing a stronger stability result for the geometric inequality $6\sqrt{3}A(K,-K)\le L(K)^2$.