Exact Uniform L1 Spacing for Solow-Polasky Diversity on Lines and Ordered Pareto Fronts

TL;DR

This paper proves that for certain diversity measures on lines and ordered sets, the optimal finite subset with fixed size is uniformly spaced, providing a natural uniform gap structure for diversity maximization.

Contribution

It establishes the uniqueness of uniform spacing as the optimal configuration for Solow-Polasky diversity on lines and ordered metric sets, extending to Pareto front approximations.

Findings

Optimal k-point subsets on [0,1] are equally spaced.

Uniform gap structure is unique for the exponential kernel.

Results extend to L1 curves and Pareto front approximations.

Abstract

We study fixed-cardinality maximization of the inverse-matrix Solow--Polasky diversity, equivalently finite metric magnitude for the exponential kernel, on one-dimensional and ordered metric sets. The analysis starts from the known finite-line gap formula for the exponential kernel, which writes the excess inverse-matrix diversity as a sum of functions of consecutive gaps. Building on this formula, the main interval theorem proves that, for every $k\geq 2$, the unique maximizing $k$-point subset of $[0,1]$ is the equally spaced set. Thus the objective selects a uniform gap representation on the real line. A converse kernel proposition shows that, among normalized non-increasing distance kernels, requiring the corresponding adjacent-gap additive structure forces the exponential family. Further results transfer the interval theorem to ordered $\ell_1$ (L1, or Manhattan) curves by isometry: the maximizing sets are uniform in accumulated $\ell_1$ length. As a consequence, monotone biobjective Pareto fronts admit Solow--Polasky optimal finite approximations that are uniformly spaced in accumulated objective-space change, a natural representation when all parts of a continuous front should be covered. Examples, including a dense connected front and a finite disconnected ZDT3 front, illustrate how the continuous uniform-gap result appears on discrete candidate sets. Solow-Polasky diversity; diversity measures; finite metric magnitude; L1 distance; uniform spacing; Pareto-front approximation; multiobjective optimization; fixed-cardinality subset selection