Extremizers and Stability of the Betke–Weil Inequality

Abstract

Let K be a compact convex domain in the Euclidean plane. The mixed area $\mathit{A}(\mathit{K},-\mathit{K})$ of K and $-\mathit{K}$ can be bounded from above by $1/(6\sqrt{3})\mathit{L}{(\mathit{K})}^2$, where $\mathit{L}(\mathit{K})$ is the perimeter of K. This was proved by Ulrich Betke and Wolfgang Weil [5]. 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 $6\sqrt{3}\mathit{A}(\mathit{K},-\mathit{K})\le \mathit{L}{(\mathit{K})}^2$.