New Characterizations of Polyhedral Cones

Abstract

A pyramid clearly has all its projections closed, even when the line segments from vertex to base are extended to infinite half-lines. Not so a circular cone. For if the cone is on its side and supported by the ( x, y ) plane in such a way that its infinite half-line of support coincides with the positive x axis, then its horizontal projection on the ( y, z ) plane is the open upper half-plane y > 0, together with the single point (0, 0).