  A well known theorem of Voronoi caracterizes extreme quadratic forms and
Euclidean lattices, that is those which are local maxima for the Hermite
function, as perfect and eutactic. This caracterization has been extended in
various cases, such that family of lattices, sections of lattices, Humbert
forms, etc. Moreover, there is a criterion for extreme lattices, discovered by
Venkov, formulated in terms of spherical designs which has been extended in the
case of Grassmannians and sections of lattices. In this article, we define a
general frame, in which there is a ``Voronoi characterization'', and a ``Venkov
criterion'' through an appropriate notion of design. This frame encompasses
many interesting situations in which a ``Voronoi characterization'' has been
proved. We also discuss the question of extremality relatively to the Epstein
zeta function, and we extend to our frame a characterization of final
zeta-extremality formulated by Delone and Ryshkov and a criterion in terms of
designs found by Coulangeon.
