  For any channel, the existence of a unique Augustin mean is established for
any positive order and probability mass function on the input set. The Augustin
mean is shown to be the unique fixed point of an operator defined in terms of
the order and the input distribution. The Augustin information is shown to be
continuously differentiable in the order. For any channel and convex constraint
set with finite Augustin capacity, the existence of a unique Augustin center
and the associated van Erven-Harremoes bound are established. The
Augustin-Legendre (A-L) information, capacity, center, and radius are
introduced and the latter three are proved to be equal to the corresponding
Renyi-Gallager quantities. The equality of the A-L capacity to the A-L radius
for arbitrary channels and the existence of a unique A-L center for channels
with finite A-L capacity are established. For all interior points of the
feasible set of cost constraints, the cost constrained Augustin capacity and
center are expressed in terms of the A-L capacity and center. Certain shift
invariant families of probabilities and certain Gaussian channels are analyzed
as examples.
