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arXiv · 2508.13722

Best approximation from the positive cone of an inner product lattice

Marouen Abdouli·Karim Boulabiar
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Abstract

Let $E$ be a directed (i.e., positively generated) ordered vector space endowed with an inner product. In this note, we prove that the following statements are equivalent: i) $E$ is a vector lattice and its norm induced by its inner product is a lattice norm. ii) The metric projection onto the positive cone $E^+$ of $E$ exists and it is both isotone and subadditive. Moreover, in this case, the best approximation to any $x \in E$ from $E^+$ coincides with its positive part $x^+$. This result extends previous work to the non-complete setting.

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Marouen Abdouli, Karim Boulabiar. 2025-08-19. Best approximation from the positive cone of an inner product lattice. https://arxiv.org/abs/2508.13722

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