Smooth Convex Bodies in R^n with Dense Union of Facets - Barov, Stoyu T. (Topology Proceedings, 2020)
Let B be closed and convex in R^n; B is called a convex body if B is compact and has a nonempty interior with respect to R^n. In addition, B is smooth if B has a unique supporting hyperplane at every boundary point. Let k, n N with k < n and let L^n_k denote the Grassmann manifold consisting of all k-dimensional linear subspaces in R^n. An intersection F of B and a supporting hyperplane is called a facet if F = n - 1. A point x of B is called exposed by P L^n_k if there is a P P such that (x + P) B = {x}. In this paper, for every n ≥ 2, we have constructed symmetric smooth convex bodies B(n) in R^n whose union of all facets is dense in the boundary of B(n) and so that the set of its facets defines a dense set P in L^n_k such that the set of all points in B(n) exposed by P is empty.