  We investigate the Mordell constant of certain families of lattices, in
particular, of lattices arising from totally real fields. We define the almost
sure value k_mu of the Mordell constant with respect to certain homogeneous
measures on the space of lattices, and establish a strict inequality k_mu <
k_nu, when mu,nu are finite and the support of mu is strictly contained in the
support of nu. In combination with known results regarding the dynamics of the
diagonal group we obtain isolation results as well as information regarding
accumulation points of the Mordell-Gruber spectrum, extending previous work of
Gruber and Ramharter. One of the main tools we develop is the associated
algebra, an algebraic invariant attached to the orbit of a lattice under a
block group, which can be used to characterize closed and finite volume orbits.
