Lebesgue covering problem's demand for convexivity

Question

Is there a specific reason why in the Lebesgue (universal) covering problem only convex sets are admitted as universal coverings? I see that for any set $X\subseteq{\bf R}^2$ of diameter $\le\alpha$, the convex hull of $X$ also has diameter $\le\alpha$ but can we also deduce that a solution to the covering problem has to be convex even if we don't demand it in the question?