  We show that the set of all possible constant diagonals of a bounded Hilbert
space operator is always convex. This, in particular, answers an open question
of J.-C. Bourin ($2003$). Moreover, we show that the joint numerical range of a
commuting operator tuple is in general not convex, which fills a gap in the
literature. We also prove that the Asplund-Ptak numerical range (which is
convex for pairs of operators) is, in general, not convex for tuples of
operators.
