Abstract
In this paper we discuss asymmetric length structures and
asymmetric metric spaces.
A length structure induces a (semi)distance function; by using
the total variation formula, a (semi)distance function induces a
length. In the first part we identify a topology in the set of paths
that best describes when the above operations are idempotent.
As a typical application, we consider the length of paths
defined by a Finslerian functional in Calculus of Variations.
In the second part we generalize the setting of General metric spaces
of Busemann, and discuss the newly found aspects of
the theory: we identify three interesting classes of paths, and
compare them; we note that a geodesic segment (as defined
by Busemann) is not necessarily continuous in our setting;
hence we present three different notions of intrinsic metric
space.
Appendix C Calculus of Variations
