  A test of the concavity of a distribution function with support contained in
the unit interval may be based on a statistic constructed from the $L^p$-norm
of the difference between an empirical distribution function and its least
concave majorant. It is shown here that the uniform distribution is least
favorable for such a test, in the sense that the limiting distribution of the
statistic obtained under uniformity stochastically dominates the limiting
distribution obtained under any other concave distribution function.
