  For any channel with a convex constraint set and finite Augustin capacity,
existence of a unique Augustin center and associated Erven-Harremoes bound are
established. Augustin-Legendre capacity, center, and radius are introduced and
proved to be equal to the corresponding Renyi-Gallager entities. Sphere packing
bounds with polynomial prefactors are derived for codes on two families of
channels: (possibly non-stationary) memoryless channels with multiple additive
cost constraints and stationary memoryless channels with convex constraints on
the empirical distribution of the input codewords.
