Linear and Sublinear Diversities
David Bryant, Paul Tupper
TL;DR
This work develops a convex-geometric framework for linear and sublinear diversities on $\mathbb{R}^k$, producing a precise integral representation $\delta(A)=\int_{\mathbb{S}^{k-1}} h_A(x)\, d\nu(x)$ for linear diversities and establishing a unique representing measure with centroid $0$. It characterizes extremal linear semidiversities as Minkowski diversities with simplex-based kernels and proves that sublinear diversities are the maximum of linear ones, akin to convex functions as maxima of linear functionals. The embedding theory is settled by showing a finite diversity embeds into a linear diversity exactly when it has negative type, and into a Minkowski (sublinear) diversity exactly when it is the maximum of negative-type diversities. Collectively, the results connect diversity theory with convex geometry and offer exact criteria for embeddability, with potential implications for hypergraph optimization and generalized geometric analysis.
Abstract
Diversities are an extension of the concept of a metric space, where a non-negative value is assigned to every finite set of points, rather than just pairs. A general theory of diversities has been developed which exhibits many deep analogies to metric space theory but also veers off in new directions. Just as many of the most important aspects of metric space theory involve metrics defined on $\Re^k$, many applications of diversity theory require a specialized theory for diversities defined on $\Re^k$, as we develop here. We focus on two fundamental classes of diversities defined on $\Re^k$: those that are Minkowski linear and those that are Minkowski sublinear. Many well-known functions in convex analysis belong to these classes, including diameter, circumradius and mean width. We derive surprising characterizations of these classes, and establish elegant connections between them. Motivated by classical results in metric geometry, and connections with combinatorial optimization, we then examine embeddability of finite diversities into $\Re^k$. We prove that a finite diversity can be embedded into a linear diversity exactly when it has negative type and that it can be embedded into a sublinear diversity exactly when it corresponds to a generalized circumradius.