Extremizers and Stability of the Betke–Weil Inequality

Bartha, Ferenc Ágoston and Bencs, Ferenc and Böröczky, Károly J. and Hug, Daniel (2022) Extremizers and Stability of the Betke–Weil Inequality. MICHIGAN MATHEMATICAL JOURNAL. pp. 1-27. ISSN 0026-2285 (In Press)

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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√ 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 √ 3A(K, −K) ≤ L(K) 2

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