The radial mean body of parameter $p>-1$ of a convex body $K \subseteq \mathbb R^n$ is a radial set $R_p K$ that was introduced by Gardner and Zhang in 1998.

They proved that if $p \geq 0$, then $R_p K$ is convex, and conjectured that this holds also for $p \in (-1, 0)$.

We prove that if $K \subseteq \mathbb R^2$ is a convex body in the plane, then $R_p K$ is convex for every $p > (-1,0)$.