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CHEBYSHEV SUBSETS OF A HILBERT SPACE SPHERE

Published online by Cambridge University Press:  21 November 2019

THEO BENDIT*
Affiliation:
University of Newcastle, Australia email [email protected]

Abstract

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.

Type
Research Article
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

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