Application of an idea of Voronoĭ to lattice zeta functions

A major problem in the geometry of numbers is the investigation of the local minima of the Epstein zeta function. In this article refined minimum properties of the Epstein zeta function and more general lattice zeta functions are studied. Using an idea of Voronoĭ, characterizations and sufficient conditions are given for lattices at which the Epstein zeta function is stationary or quadratic minimum. Similar problems of a duality character are investigated for the product of the Epstein zeta function of a lattice and the Epstein zeta function of the polar lattice. Besides Voronoĭ type notions such as versions of perfection and eutaxy, these results involve spherical designs and automorphism groups of lattices. Several results are extended to more general lattice zeta functions, where the Euclidean norm is replaced by a smooth norm.