  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 $1/(6\sqrt{3})L(K)^2$,
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 $6\sqrt{3}A(K,-K)\le L(K)^2$.
