Voronoi's idea, to identify quadratic forms in E^d and points in E^d(d+1)/2, has led to many results on positive definite quadratic forms, zeta functions, normed spaces, etc. In this lecture we study refined extremum properties of the density of lattice packings of balls and of Epstein zeta functions, related to results of Berge, Martinet, Bachoc, Coulangeon, Venkov and others.