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  "path": "/abs/2605.21922v1",
  "publishedAt": "2026-05-22T00:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Michael T. M. Emmerich",
    "Mahboubeh Nezhadmoghaddam",
    "Jesús Guillermo Falcón Cardona"
  ],
  "textContent": "**Authors:** Michael T. M. Emmerich, Mahboubeh Nezhadmoghaddam, Jesús Guillermo Falcón Cardona\n\nWe 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",
  "title": "Exact Uniform L1 Spacing for Solow-Polasky Diversity on Lines and Ordered Pareto Fronts"
}