Cover and bounded parallel faces (Q1275313)

\textit{D. Gale} and \textit{V. Klee} [Math. Scand. 7, 379--391 (1960; Zbl 0115.16501)] showed that the conical hull of a ``continuous'' convex set from a point which does not belong to this convex set is closed. In 1962, \textit{C. C. Braunschweiger} and \textit{H. E. Clark} [Am. Math. Mon. 69, 272--277 (1962; Zbl 0116.08302)] asked the question: when is the conical hull \(\mathbb C(A,x)\) of a closed convex set \(A\) in \(\mathbb R^n\) from a point \(x\) closed? This problem was solved in 1983 by \textit{J. Bair} and \textit{F. Jongmans} [Bull. Soc. R. Sci. Liège 52, 285--294 (1983; Zbl 0526.52001)]: these authors proved the theorem of the closed conical hull by introducing the notion of cover. At first, the cover \(\mathbb G(A)\) of a closed convex set \(A\) was defined as the intersection of some ``support-cones'' of \(A\) from marginal rays and asymptotes. Now, several equivalent definitions are given for the cover, especially, the cover is the shadow from the recession cone. The authors use the cover or the closure of the cover in order to determine when there exists a translate \(H'\) (resp. \(H''\)) of the hyperplane \(H\) such that its intersection with a closed convex set \(A\) (resp. \(B\)) is a bounded face. It is shown that the study of the closure of the cover gives interesting results about the cover itself.