Best approximations in normed vector spaces

Abstract

We investigate which subsets of normed vector spaces are Chebyshev, that is, they admit a unique best approximation for every vector. We show that a subset of a strictly convex uniformly smooth finite-dimensional normed vector space is Chebyshev if, and only if, it is non-empty closed and convex. We also show that any non-empty closed convex subset of a strictly convex reflexive normed vector space is Chebyshev. We finally take a look at a few examples of applicable normed vector spaces and a few counter examples to some intuitions one might have.