Planar radial mean bodies are convex

The radial mean bodies of parameter $p>-1$ of a convex body $K \subseteq \mathbb R^n$ are radial sets introduced in [4] by Gardner and Zhang. They are known to be convex for $p\geq 0$. We prove that if $K \subseteq \mathbb R^2$ is a convex body, then its radial mean body of parameter $p$ is convex for every $p \in (-1,0)$.
Comment: comments are welcome