Augustin Information in the Vicinity of Augustin Capacity-Achieving Input Distributions

For channels with finite input and output sets, under mild technical assumptions, the local behavior of the Augustin information as a function of the input distribution is characterized for all positive orders using the implicit function theorem and the characterization of the Augustin information in terms of the Augustin dual. For channels with (potentially multiple) linear constraints, the slowest decrease of Augustin information with increasing distance from the Augustin capacity-achieving input distributions is characterized within small neighborhoods around these distributions for all positive orders.