The Varchenko determinant of an oriented matroid

Varchenko introduced in ۱۹۹۳ 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 ۲۰۱۷, 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\"{a}ttler and Welker also computed in ۲۰۱۹ the same determinant.