Directional Fineness for Radiant and Co-Radiant Sets: Separation and Quantitative Approximation
Abstract
A prescribed directionally fine family of cones yields separation and primal bipolar identities for closed radiant and co-radiant sets in real normed spaces. Positive levels of adapted positively homogeneous functions separate points from closed radiant sets, while a correction by the norm provides bounded radiant neighbourhoods for co-radiant separation. The resulting classes also calibrate radial thresholds and represent gauges and cogauges through strict separators. For convex directionally fine families, the separating generators can be chosen Lipschitz and superlinear, yielding convex separating superlevels. Closed co-radiant sets with compact direction sets admit finite relative outer approximations by positive superlevels of selected generators, with separation preservation and directional and bounded-section error estimates, without requiring continuity or boundedness of the radial profile. Superlinear generators yield a global residual error bound, exact penalization for Lipschitz objectives, recovery estimates, and convergence of bounded approximate minimizers. An explicit convex directionally fine axial family is available in every real normed space. Its norm-corrected functions realize separation and bipolarity without rotundity assumptions, while an axial recovery map improves the relative distance factor from η/(1−η) to η. A three-ray production model illustrates the quantitative recovery estimates.
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BibTeX
@misc{garciacastano2026directional,
title = {{Directional Fineness for Radiant and Co-Radiant Sets: Separation and Quantitative Approximation}},
author = {Fernando García-Castaño and Miguel Ángel Melguizo-Padial},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25141},
eprint = {2610.25141},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25141},
}
RIS
TY - UNPB
TI - Directional Fineness for Radiant and Co-Radiant Sets: Separation and Quantitative Approximation
AU - Fernando García-Castaño
AU - Miguel Ángel Melguizo-Padial
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25141
UR - https://arxiv.org/abs/2610.25141
AB - A prescribed directionally fine family of cones yields separation and primal bipolar identities for closed radiant and co-radiant sets in real normed spaces. Positive levels of adapted positively homogeneous functions separate points from closed radiant sets, while a correction by the norm provides bounded radiant neighbourhoods for co-radiant separation. The resulting classes also calibrate radial thresholds and represent gauges and cogauges through strict separators. For convex directionally fine families, the separating generators can be chosen Lipschitz and superlinear, yielding convex separating superlevels. Closed co-radiant sets with compact direction sets admit finite relative outer approximations by positive superlevels of selected generators, with separation preservation and directional and bounded-section error estimates, without requiring continuity or boundedness of the radial profile. Superlinear generators yield a global residual error bound, exact penalization for Lipschitz objectives, recovery estimates, and convergence of bounded approximate minimizers. An explicit convex directionally fine axial family is available in every real normed space. Its norm-corrected functions realize separation and bipolarity without rotundity assumptions, while an axial recovery map improves the relative distance factor from η/(1−η) to η. A three-ray production model illustrates the quantitative recovery estimates.
ER -