  This paper is concerned with the problem of determining whether a
projective-equivalence class of sprays is the geodesic class of a Finsler
function. We address both the local and the global aspects of this problem. We
present our results entirely in terms of a multiplier, that is, a type (0,2)
tensor field along the tangent bundle projection. In the course of the analysis
we consider several related issues of interest including the positivity and
strong convexity of positively-homogeneous functions, the relation to the
so-called Rapcs\'ak conditions, some peculiarities of the two-dimensional case,
and geodesic convexity for sprays.
