Convex analysis in normed spaces and metric projections onto convex bodies
Vitor Balestro, Horst Martini, Ralph Teixeira

TL;DR
This paper explores the properties of metric projections and distance functions in finite-dimensional normed spaces with convex bodies, using Legendre transforms to provide new insights and relations between convex functions and geometric properties.
Contribution
It introduces a novel approach by identifying the vector space with its dual via Legendre transforms, offering new interpretations of convex functions and their geometric relations.
Findings
New relations between convex functions and geometric properties of norms
Re-interpretations of properties of convex functions in normed spaces
Enhanced understanding of metric projections onto convex bodies
Abstract
We investigate metric projections and distance functions referring to convex bodies in finite-dimensional normed spaces. For this purpose we identify the vector space with its dual space by using, instead of the usual identification via the standard inner product, the Legendre transform associated with the given norm. This approach yields re-interpretations of various properties of convex functions, and new relations between such functions and geometric properties of the studied norm are also derived.
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Taxonomy
TopicsOptimization and Variational Analysis · Fixed Point Theorems Analysis · Point processes and geometric inequalities
