{"ID":2873306,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.06392","arxiv_id":"2509.06392","title":"What are Capra-Convex Sets?","abstract":"This paper focuses on a specific form of abstract convexity known as Capra-convexity, where a constant along primal rays (Capra) coupling replaces the scalar product used in standard convex analysis to define generalized Fenchel conjugacies. A key motivating result is that the ${\\ell}$0 pseudonorm - which counts the number of nonzero components in a vector - is equal to its Capra-biconjugate. This implies that ${\\ell}$0 is a Capra-convex function, highlighting potential applications in statistics and machine learning, particularly for enforcing sparsity in models. Building on prior work characterizing the Capra-subdifferential of ${\\ell}$0 and the role of source norms in defining the Capra-coupling, the paper provides a characterization of Capra-convex sets.","short_abstract":"This paper focuses on a specific form of abstract convexity known as Capra-convexity, where a constant along primal rays (Capra) coupling replaces the scalar product used in standard convex analysis to define generalized Fenchel conjugacies. A key motivating result is that the ${\\ell}$0 pseudonorm - which counts the nu...","url_abs":"https://arxiv.org/abs/2509.06392","url_pdf":"https://arxiv.org/pdf/2509.06392v2","authors":"[\"Jean-Philippe Chancelier\",\"Michel de Lara\",\"Adrien Le Franc\",\"Seta Rakotomandimby\"]","published":"2025-09-08T07:23:48Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}