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](https://mathoverflow.net/questions/157587/2f0-sup-z-w-in-d-fz-fw-if-and-only-if-f-is-linear/157616#157616)



Maximal regions with given diameter
I am pretty sure that the condition you state is equivalent to $D$ being of constant width, not just in the plane, but in every dimension. See, for instance <http://www.ciem.unican.es/encuentros/banach/2012/moreno1.pdf>


Other references:


Dalla, Leoni; Tamvakis, N. K. Sets of constant width and diametrically complete sets in normed spaces. Bull. Soc. Math. Grèce (N.S.) 26 (1985), 27–39, MR0854917.


Moreno, José Pedro; Schneider, Rolf Diametrically complete sets in Minkowski spaces. Israel J. Math. 191 (2012), no. 2, 701–720, MR3011492.


