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arXiv · 2509.06392

What are Capra-Convex Sets?

Jean-Philippe Chancelier·Michel de Lara·Adrien Le Franc·Seta Rakotomandimby
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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.

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Jean-Philippe Chancelier, Michel de Lara, Adrien Le Franc, Seta Rakotomandimby. 2025-09-08. What are Capra-Convex Sets?. https://arxiv.org/abs/2509.06392

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