  Varchenko introduced in 1993 a distance function on the chambers of a
hyperplane arrangement that gave rise to a determinant whose entry in position
$(C, D)$ is the distance between the chambers $C$ and $D$, and computed that
determinant. In 2017, Aguiar and Mahajan provided a generalization of that
distance function, and computed the corresponding determinant. This article
extends their distance function to the topes of an oriented matroid, and
computes the determinant thus defined. Oriented matroids have the nice property
to be abstractions of some mathematical structures including hyperplane and
sphere arrangements, polytopes, directed graphs, and even chirality in
molecular chemistry. Independently and with another method, Hochst\"attler and
Welker also computed in 2019 the same determinant.

Combinatorics