Relation between metric spaces and Finsler spaces

In a connected Finsler space Fn=(M,F)Fn=(M,F) every ordered pair of points p,q∈Mp,q∈M determines a distance ϱF(p,q)ϱF(p,q) as the infimum of the arc length of curves joining p to q  . (M,ϱF)(M,ϱF) is a metric space if FnFn is absolutely homogeneous, and it is quasi-metric space (i.e. the symmetry: ϱF(p,q)=ϱF(q,p)ϱF(p,q)=ϱF(q,p) fails) if FnFn is positively homogeneous only. It is known the Busemann–Mayer relation limt→t0+ddtϱF(p0,p(t))=F(p0,p˙0), for any differentiable curve p(t)p(t) in an FnFn. This establishes a 1:11:1 relation between Finsler spaces Fn=(M,F)Fn=(M,F) and (quasi-) metric spaces (M,ϱF)(M,ϱF).We show that a distance function ϱ(p,q)ϱ(p,q) (with the differentiability property of ϱFϱF) needs not to be a ϱFϱF. This means that the family {(M,ϱ)}{(M,ϱ)} is wider than {(M,ϱF)}{(M,ϱF)}. We give a necessary and sufficient condition in two versions for a ϱ   to be a ϱFϱF, i.e. for a (quasi-) metric space (M,ϱ)(M,ϱ) to be equivalent (with respect to the distance) to a Finsler space (M,F)(M,F).