  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.
