Metric projections in inner product spaces

Given a subset 𝐶 of a metric space 𝑀, and a point 𝑥 ∈ 𝑀, the metric projection 𝑃_𝐶(𝑥) of 𝑥 onto 𝐶 is defined as the set of points in 𝐶 of minimal distance from 𝑥; the set of best approximations of 𝑥 by points in 𝐶. This set-valued map is of clear and fundamental interest in approximation theory. Though simple to define, metric projections can have surprising complexity, even in well-structured spaces, such as Hilbert Spaces, making them a topic of abstract interest as well. A particularly interesting class of sets are Chebyshev Sets, sets whose metric projection maps are everywhere single-valued. Bunt proved in 1934 that such sets in finitedimensional Euclidean space are convex, while in 1958, Efimov and Stečkin conjectured the same might be true in general real Hilbert Spaces, or even reflexive smooth spaces. This conjecture came to be known as the Chebyshev Conjecture. Despite much research into Chebyshev sets since, the conjecture remains open today. Chapter 1 builds the theory of Chebyshev sets, reviewing notable results about Chebyshev sets both in finite dimensions and in real Hilbert spaces. In Chapter 2, we first examine uniquely remotal sets, subsets of metric spaces with unique furthest points. In the context of real inner product spaces, an important result by Asplund, Ficken and Klee connects the characterisation of Chebyshev sets with the characterisation of uniquely remotal sets, using geometric properties of inversion in the sphere maps. A stronger, local version of this theorem is provided. We then extend the theorem, using instead the analogous properties of the stereographic projection map. This leads to a natural extension of Chebyshev and uniquely remotal sets, a condition which we call Sphere-Chebyshev sets. Such sets have superior structure and more symmetries than both Chebyshev and unqiuely remotal sets. Characterisation theorems are proven both for Sphere-Chebyshev sets, and a notable group of their symmetries, called Generalised Möbius Transformations. Finally, in Chapter 3, we develop a characterisation of closed subsets 𝐶 in real Hilbert spaces 𝑋, which fail to admit a point 𝑥 ∈ 𝑋 such that |𝑃_𝐶(𝑥)| = 2 and 𝑃_𝐶 is uppersemicontinuous at 𝑥. We show this property to be equivalent both to a stronger connectedness property known as 𝐵∘-connectedness, as well as a local structural property which we call Locally-determined Negative Set Curvature. This result is then used to provide some necessary local geometric conditions for a set to be Chebyshev.