Shadows of Cube Vertices

Abstract

Suppose that a finite-dimensional cube is orthogonally projected onto a central section of itself by a subspace of one dimension less. Up to dimension $9$, at least one vertex is projected onto the section, but for dimension $10$ or larger, there are orthogonal projections for which all the vertices are projected outside the section. In fact, this is the case for "most" orthogonal projections, as the dimension tends to infinity.