AbstractGiven ann-dimensional hypercubeC(n), is there a hyperplane that meets the interior of every facet ofC(n)? Contrary to the impression given by the casesn= 1 andn= 2, the answer is“yes”for alln>2. Furthermore, forn>3, the hyperplane can also be chosen to contain any prescribed point ofC(n). Such results imply that the center of the smallest hyper-rectangle having sides parallel to the coordinate planes and enclosing a convex polytopeKdoes not necessarily lie inK. A practical estimation problem motivating the inquiry is described.