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Abstract
In this thesis we study certain roundness inequalities in metric spaces. The properties roundness and generalised roundness of metric spaces were originally introduced by Enflo where they were used to act as obstructions to uniform embeddings. Since then the relation of these properties to other embeddings such as isometric embeddings and coarse embeddings of metric spaces has been the subject of much study. A major result in this area is that generalised roundness, and the equivalent property of $p$-negative type, can act as a sufficient condition for certain isometric embeddings into Euclidean space. In particular a finite metric space $(X,d_{X})$ embeds isometrically into some Euclidean space if and only if it has $2$-negative type.
We start by studying roundness in the setting of Banach spaces. Here we are able to unify and expand upon results of Enflo that aid in the calculation of the maximal roundness of many classical Banach spaces. We then consider the problem of computing the maximal roundness of more general Banach spaces whose unit spheres are easy to visualise but whose norm is more complicated to write down. In doing so we are led naturally to a slightly broader class of inequalities than that of the usual roundness. This new class of inequalities is then shown to have a close connection to the geometric concepts of smoothness and convexity.
In the next section of this thesis we investigate the possibility of an analogous class of inequalities in the setting of generalised roundness, or equivalently $p$-negative type. Our starting point is a theorem of Linial, London and Rabinovich which characterises those finite metric spaces that admit a bi-lipschitz embedding into some Euclidean space with a given amount of distortion. Using this we are able to define the new concept of distorted $p$-negative type which we then show is a generalisation of the usual $p$-negative type. Due to the theorem of Linial, London and Rabinovich the concept of distorted $p$-negative type acts a sufficient condition for certain bi-lipschitz embeddings into Euclidean space. In particular we show that a finite metric space $(X,d_{X})$ admits a bi-lipschitz embedding into some Euclidean space with distortion at most $C$ if and only if it has $2$-negative type with distortion $C$. We are also able to prove an analogous result for infinite metric spaces. We then proceed to generalise the properties of the usual $p$-negative type such as strictness and polygonal equalities to this distorted setting. Explicit examples of finite metric spaces with distorted $p$-negative type and examples of their distorted polygonal equalities are then given. Finally, we prove a certain linearisation of the theorem of Linial, London and Rabinovich which provides a complete characterisation of those Banach spaces that are linearly isomorphic to a Hilbert space with a given level of distortion.
