On the volume of convolution bodies in the plane

For every convex body $K\subset {\mathbb{R}}^n$ and $\delta \in (0,1)$, the $\delta$-convolution body of $K$ is the set of $x\in {\mathbb{R}}^n$ for which ${\mid K\cap (K+x)\mid }_n\ge \delta {\mid K\mid }_n$. We show that for $n=2$ and any $\delta \in (0,1)$, ellipsoids do not maximize the volume of the $\delta$-convolution body of $K$, when $K$ runs over all convex bodies of a fixed volume. This behavior is somehow unexpected and contradicts the limit case $\delta \to 1^{-}$, which is governed by the Petty projection inequality.