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arXiv · 2103.11672

Extremizers and stability of the Betke--Weil inequality

Ferenc A. Bartha·Ferenc Bencs·Károly J. Böröczky·Daniel Hug
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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 $6\sqrt{3}A(K,-K)\le L(K)^2$.

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Ferenc A. Bartha, Ferenc Bencs, Károly J. Böröczky, Daniel Hug. 2021-03-22. Extremizers and stability of the Betke--Weil inequality. https://arxiv.org/abs/2103.11672

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