Finslerian geometry in low regularity

A classical result from  1941 by H. Busemann and W. Mayer  states that a Finslerian structure on a variety is determined by the associated distance function.  Unfortunately, Busemann-Mayer's original article is difficult to read and the proof never seems to have been the subject of a more modern and/or more pedagogical reformulation. The aim of this presentation will be to revisit Busemann-Mayer's theorem and to  relate it to contemporary research in metric and Finslerian geometry of low regularity. In particular, we will prove that the convexification of a semi-continuous pre-Finslerian metric induces the same distance as the pre-Finslerian metric itself. We will also show  some results on the metric derivative and the regularity of the minimizing curves for a Finslerian metric of low regularity.