Marshall's lemma for convex density estimation

Marshall's [Nonparametric Techniques in Statistical Inference (1970) 174--176] lemma is an analytical result which implies $\sqrt{n}$--consistency of the distribution function corresponding to the Grenander [Skand. Aktuarietidskr. 39 (1956) 125--153] estimator of a non-decreasing probability density. The present paper derives analogous results for the setting of convex densities on $[0,\infty)$.