  We introduce a new generalization of relative entropy to non-negative vectors
with sums $\gt 1$. We show in a purely combinatorial setting, with no
probabilistic considerations, that in the presence of linear constraints
defining a convex polytope, a concentration phenomenon arises for this
generalized relative entropy, and we quantify the concentration precisely. We
also present a probabilistic formulation, and extend the concentration results
to it. In addition, we provide a number of simplifications and improvements to
our previous work, notably in dualizing the optimization problem, in the
concentration with respect to $\ell_{\infty}$ distance, and in the relationship
to generalized KL-divergence. A number of our results apply to general compact
convex sets, not necessarily polyhedral.
