Hyperplanes That Intersect Each Ray of a Cone Once and a Banach Space Counterexample

Abstract

Suppose $C$ is a cone contained in real vector space $V$. When does $V$ contain a hyperplane $H$ that intersects each of the 0-rays in $C\setminus \{\mathrm{0}\}$ exactly once? We build on results found in Aliprantis, Tourky, and Klee Jr.’s work to give a partial answer to this question. We also present an example of a salient, closed Banach space cone $C$ for which there does not exist a hyperplane that intersects each 0-ray in $C\setminus \{\mathrm{0}\}$ exactly once.