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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

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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.

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Thanks for the nice reference. How do you show that my condition is equivalent to constant width? –  Alexandre Eremenko Feb 16 at 20:55
    
I think the 1985 reference contains a proof, or at least a reference to an earlier article with a proof. The proof is not too hard, as I recall. –  Wlodek Kuperberg Feb 16 at 21:12
    
OK, thanks. Unfortunatey this reference is not easily available. –  Alexandre Eremenko Feb 16 at 21:23
    
A couple of details in the proof: (1) A body of constant width is convex; (2) A convex body is of constant width if and only if it has a diameter parallel to every direction. The latter condition is equivalent to being maximal with respect to diameter. –  Wlodek Kuperberg Feb 16 at 21:26

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