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

Abstract EN

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∖{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∖{0} exactly once.