  We review various characterizations of uniform convexity and smoothness on
norm balls in finite-dimensional spaces and connect results stemming from the
geometry of Banach spaces with \textit{scaling inequalities} used in analysing
the convergence of optimization methods. In particular, we establish local
versions of these conditions to provide sharper insights on a recent body of
complexity results in learning theory, online learning, or offline
optimization, which rely on the strong convexity of the feasible set. While
they have a significant impact on complexity, these strong convexity or uniform
convexity properties of feasible sets are not exploited as thoroughly as their
functional counterparts, and this work is an effort to correct this imbalance.
We conclude with some practical examples in optimization and machine learning
where leveraging these conditions and localized assumptions lead to new
complexity results.
