This paper examines the efficient estimation of partially identified models defined by moment inequalities that are convex in the parameter of interest. In such a setting, the identified set is itself convex and hence fully characterized by its support function. We provide conditions under which, despite being an infinite dimensional parameter, the support function admits √‐consistent regular estimators. A semiparametric efficiency bound is then derived for its estimation, and it is shown that any regular estimator attaining it must also minimize a wide class of asymptotic loss functions. In addition, we show that the “plug‐in” estimator is efficient, and devise a consistent bootstrap procedure for estimating its limiting distribution. The setting we examine is related to an incomplete linear model studied in Beresteanu and Molinari (2008) and Bontemps, Magnac, and Maurin (2012), which further enables us to establish the semiparametric efficiency of their proposed estimators for that problem.