On the Structure of the Complements of Chebyshev Sets

A set is called a Chebyshev set if it contains a unique best approximation element. We study the structure of the complements of Chebyshev sets, in particular considering the following question: How many connected components can the complement of a Chebyshev set in a finite-dimensional normed or nonsymmetrically normed linear space have? We extend some results from [A. R. Alimov, East J. Approx, 2, No. 2, 215–232 (1996)]. A. L. Brown's characterization of four-dimensional normed linear spaces in which every Chebyshev set is convex is extended to the nonsymmetric setting. A characterization of finite-dimensional spaces that contain a strict sun whose complement has a given number of connected components is established.