Multifractal analysis via Lagrange duality

We provide a self-contained exposition of the well-known multifractal formalism for self-similar measures satisfying the strong separation condition. At the heart of our method lies a pair of quasiconvex optimization problems which encode the parametric geometry of the Lagrange dual associated with the constrained variational principle. We also give a direct derivation of the Hausdorff dimension of the level sets of the upper and lower local dimensions by exploiting certain weak uniformity properties of the space of Bernoulli measures.