  We formulate an affine invariant implementation of the accelerated
first-order algorithm in Nesterov (1983). Its complexity bound is proportional
to an affine invariant regularity constant defined with respect to the
Minkowski gauge of the feasible set. We extend these results to more general
problems, optimizing H\"older smooth functions using $p$-uniformly convex prox
terms, and derive an algorithm whose complexity better fits the geometry of the
feasible set and adapts to both the best H\"older smoothness parameter and the
best gradient Lipschitz constant. Finally, we detail matching complexity lower
bounds when the feasible set is an $\ell_p$ ball. In this setting, our upper
bounds on iteration complexity for the algorithm in Nesterov (1983) are thus
optimal in terms of target precision, smoothness and problem dimension.
