Hyperplanes that intersect each ray of a cone once and a Banach space counterexample. (English) Zbl 1470.52003

McCarthy, Chris

Abstr. Appl. Anal. 2016, Article ID 9623090, 7 p. (2016).

Summary: 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 \(C\, \backslash \{0 \}\) exactly once? We build on results found in [C. D. Aliprantis and R. Tourky, Cones and duality. Providence, RI: American Mathematical Society (AMS) (2007; Zbl 1127.46002)] and Klee jun.’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 \(C\, \backslash \{0 \}\) exactly once.

MSC:

52A07	Convex sets in topological vector spaces (aspects of convex geometry)
46A32	Spaces of linear operators; topological tensor products; approximation properties
47L07	Convex sets and cones of operators

Citations:

Zbl 1127.46002

References:

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