Introduction

One of the most famous questions of functional and numerical analysis is the best approximation problem: find the nearest point (also called the projection) from a given point to a nonempty set, both situated in an ambient space endowed with a certain metric structure by means of which one can measure metric distances. In the classical theory of best approximations, the ambient space has a vector space structure, and the distance function is symmetric, which comes from a norm or, in particular from an inner product. However, nonsymmetry is abundant in important real life situations; indeed, it is enough to consider swimming (with/against a current) in a river, or walking (up/down) a mountain slope. These kinds of nonsymmetric phenomena lead us to spaces with no possible vector space structure, while the distance function is not necessarily symmetric; see also Chapter 14. Thus, the most appropriate framework is to consider not necessarily reversible Finsler manifolds. This nonlinear context throws completely new light upon the problem of best approximations where well-tried methods usually fail.

The purpose of this chapter is to initiate a systematic study of best approximation problems on Finsler manifolds by exploiting notions from Finsler geometry as geodesics, forward/backward geodesically completeness, flag curvature, fundamental inequality of Finsler geometry, and first variation of arc length.