On Constant Width Sets in Hilbert Spaces and Around

We present some results concerning the family

of sets with constant width and the family

\mathcal{DM} D M

of diametrically maximal (or diametrically complete) sets. In particular we prove (Theorem 2) that

\mathcal{CW} = \mathcal{DM} C W = D M

in Hilbert spaces.