2021-03-22
Extremizers and stability of the Betke--Weil inequality
Publication
Publication
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/(63–√)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 63–√A(K,−K)≤L(K)2.
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Bartha, F. A., Bencs, F., Böröczky, K. J., & Hug, D. (2021). Extremizers and stability of the Betke--Weil inequality. |
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