Type One Functions and Voronoï's Theorem

Voronoï's theorem characterizes local maxima of the Hermite invariant m / det^1/n defined on the open cone of positive definite n by n symmetric matrices, where m denotes the arithmetical minimum function. In this paper, we extend Voronoï's theorem to functions of the form m / φ when φ is a type one function. Moreover, we study the Hermite like constant defined from m / φ