Let us call a bounded region $D$ in the plane maximal if the conditions $D\subset D'$ and $\mathrm{diam} D'=\mathrm{diam} D$ imply $D'=D$. Is it possible to describe all maximal regions? The only examples I know are discs and Reuleaux triangles.

If a complete description is difficult, can one prove some properties of maximal regions? For example, I suppose that they must be convex. Do they have piecewise-analytic boundaries?

This is inspired by my answer to another MO question