Linear and Sublinear Diversities

Diversities are an extension of the concept of a metric space which assign a non-negative value 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 $\mathbb{R}^k$, many applications of diversity theory require a specialized theory for diversities defined on $\mathbb{R}^k$, as we develop here. We focus on two fundamental classes of diversities defined on $\mathbb{R}^k$: 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 $\mathbb{R}^k$. We prove that a finite diversity can be embedded into a linear diversity exactly when it is of negative type and that it can be embedded into a sublinear diversity exactly when it corresponds to a generalized circumradius.

💡 Research Summary

The paper develops a specialized theory of diversities—set‑valued extensions of metrics—on Euclidean spaces ℝ^k, focusing on two fundamental families: Minkowski linear diversities and Minkowski sublinear diversities. A diversity δ assigns a non‑negative value to every finite subset A of a ground set X and satisfies natural axioms (non‑negativity, monotonicity, and a set‑based triangle inequality). When restricted to pairs, δ yields the induced metric d(x,y)=δ({x,y}).

The authors first collect canonical examples. The ℓ₁‑diversity δ₁(A)=∑i max{a,b∈A}|a_i−b_i| is linear; the diameter diversity induced by any norm is sublinear; the circumradius (or generalized circumradius) diversity δ_K(A)=inf{λ≥0 : A⊆λK+x} for a compact convex kernel K is sublinear but not linear in general; the mean‑width diversity is linear; and the zonotope diversity (minimum length of a zonotope containing A) is sublinear. These examples illustrate that linear diversities behave like norms on the space of finite sets, while sublinear diversities satisfy only a subadditivity condition.

A central technical contribution is a complete representation theorem for linear diversities (Theorem 5). The authors prove that any linear (or linear semidiversity) δ can be written as an integral of support functions over the unit sphere: \