Convex Analysis in Normed Spaces and Metric Projections onto Convex Bodies

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. Here the concept of Birkhoff orthogonality, mostly common in finite-dimensional normed spaces, will play an important role.