Abstract:
A Chebyshev set is a subset of a normed linear space that admits unique best approximations. In the first part of this thesis we focus on some positive results concerning Chebyshev sets, deriving properties of the metric projection, sufficient conditions for a subset of a normed linear space to be a Chebyshev set, and sufficient conditions for a Chebyshev set to be convex. In the second half of the thesis we look at the so called ‘Chebyshev set problem’, constructing a highly non-trivial example of a non-convex Chebyshev set in an inner product space.