Convex analysis in normed spaces and metric projections onto convex bodies

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.