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.