Extremizers and Stability of the Betke–Weil Inequality

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 [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 3A(K,−K)≤L(K)2.