The Euler beta-function, the Vandermonde determinant, Legendre's equation, and critical values of linear functions on a configuration of hyperplanes. I

When working with a set of linear functions and the configration of hyperplanes they define, one encounters a matrix of multidimensional hypergeometric integrals. We give a formula which expresses the determinant of this matrix in terms of critical values of the linear functions restricted to the configuration.
Bibliography: 26 titles.