Convex analysis in normed spaces and metric projections onto convex bodies

V. Balestro, H. Martini and R. Teixeira, Convex analysis in normed spaces and metric projections onto convex bodies, 2019.

Comments: We were inspired by the following question: assume that a finite-dimensional vector space is endowed with a norm which is smooth and strictly convex, but is not necessarily induced by an inner product, and let K be a fixed convex body. Is the distance function (in the norm) to K differentiable? Furthermore, how can one characterize its gradient? We answer positively to the first question, actually proving C^1 regularity, and we also give a geometric description of the gradient of a distance function. To do so, we relate the theory of convex functions to the geometry given by norms on their respective domains. By identifying the original vector space and its dual via the Legendre transform, we get new concepts of gradient and sub-gradient, and we show how these are related to the geometry given by the norm in the sub-level sets.