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Directionally Fine Families of Cones: Construction Principles, Approximation, and Separation — arXiv:2609.15337

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.15337
License (as archived): CC0-1.0
Archived: 2026-10-16T00:39:21.226Z
SHA-256 of the archived text: 42b77421d269ca8ac69018c3c2045b543e32ebff20a0a2a4cffcddd5de5c1e2a
Stable URL of this page: https://metraipolis.com/documento/dc_a79fc8dcdec0

How to cite

Directionally Fine Families of Cones: Construction Principles, Approximation, and Separation — arXiv:2609.15337 — arXiv (en), https://arxiv.org/abs/2609.15337 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_a79fc8dcdec0 (Metraipolis archive dc_a79fc8dcdec0, sha256 42b77421d269ca8ac69018c3c2045b543e32ebff20a0a2a4cffcddd5de5c1e2a)

BibTeX

@misc{dca79fc8dcdec0,
  title = {Directionally Fine Families of Cones: Construction Principles, Approximation, and Separation — arXiv:2609.15337},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.15337},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_a79fc8dcdec0; sha256 42b77421d269ca8ac69018c3c2045b543e32ebff20a0a2a4cffcddd5de5c1e2a},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.15337. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We introduce the notion of a directionally fine family of cones, a local geometric property expressing the possibility of selecting arbitrarily narrow cones around every direction of a normed space. We show that this local property provides a common mechanism for finite separation and approximation. In particular, directionally compact sets can be separated from closed cones by finitely many members of the family, while locally compact cones admit finite outer approximations with quantitative Hausdorff control on their sets of directions and on bounded sections. These finite constructions lead naturally to max-type positively homogeneous separators and do not require convexity of the cones to be separated. We then study two complementary realizations of the framework. For the classical family of Bishop--Phelps cones, directional fineness is characterized by the requirement that every point of the unit sphere be a denting point of the unit ball, equivalently, by property~$(G)$; in Banach spaces this amounts to rotundity together with the Kadec property. On the other hand, we introduce transversal coercivity as a general axial construction principle. It yields directionally fine families in arbitrary normed spaces, without geometric assumptions on the norm, and includes uniform axial deviation and norm-normalized axial cones as canonical models. Finally, the explicit structure of the former yields penalization and recovery results for optimization over locally compact cones, together with quantitative approximation and convergence estimates.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-15 in math.OC).
Categorie arXiv: Optimization and Control (math.OC).
Pagina del lavoro: https://arxiv.org/abs/2609.15337 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-16T00:16:48.838Z dalla pagina https://arxiv.org/catchup/math.OC/2026-09-15?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).