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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.