Closed convex sets with an open or closed Gauss range

We characterize the closed convex subsets of Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^{n}$$\end{document} which have open or closed Gauss ranges. Some special attention is paid to epigraphs of lower semicontinuous convex functions.