We prove a second Marshall inequality for adaptive convex density estimation via least squares. The result completes the
first inequality proved recently by Du¨ mbgen et al. [2007. Marshall’s lemma for convex density estimation. IMS Lecture
Notes—Monograph Series, submitted for publication. Preprint available at hhttp://arxiv.org/abs/math.ST/0609277i], and
is very similar to the original Marshall inequality in monotone estimation.