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Chebyshev subsets of a Hilbert space sphere. (English) Zbl 1428.41033

Summary: The Chebyshev conjecture posits that Chebyshev subsets of a real Hilbert space \(X\) are convex. Works by Asplund, Ficken and Klee have uncovered an equivalent formulation of the Chebyshev conjecture in terms of uniquely remotal subsets of \(X\). In this tradition, we develop another equivalent formulation in terms of Chebyshev subsets of the unit sphere of \(X\times \mathbb{R} \). We characterise such sets in terms of the image under stereographic projection. Such sets have superior structure to Chebyshev sets and uniquely remotal sets.

MSC:

41A50 Best approximation, Chebyshev systems
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
46B20 Geometry and structure of normed linear spaces
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