Extremizers and Stability of the Betke-Weil Inequality

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 root 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 root 3A(K, -K) <= L(K)(2).