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Le Cam's method (or the two-point method) is a commonly used tool forobtaining statistical lower bound and especially popular for functionalestimation problems. This work aims to explain and give conditions for thetightness of Le Cam's lower bound in functional estimation from the perspectiveof convex duality. Under a variety of settings it is shown that themaximization problem that searches for the best two-point lower bound, upondualizing, becomes a minimization problem that optimizes the bias-variancetradeoff among a family of estimators.  For estimating linear functionals of a distribution our work strengthensprior results of Donoho-Liu \cite{DL91} (for quadratic loss) by dropping theH\"olderian assumption on the modulus of continuity. For exponential familiesour results extend those of Juditsky-Nemirovski \cite{JN09} by characterizingthe minimax risk for the quadratic loss under weaker assumptions on theexponential family.  We also provide an extension to the high-dimensional setting for estimatingseparable functionals. Notably, coupled with tools from complex analysis, thismethod is particularly effective for characterizing the ``elbow effect'' -- thephase transition from parametric to nonparametric rates. As the mainapplication we derive sharp minimax rates in the Distinct elements problem(given a fraction $p$ of colored balls from an urn containing $d$ balls, theoptimal error of estimating the number of distinct colors is $\tilde\Theta(d^{-\frac{1}{2}\min\{\frac{p}{1-p},1\}})$) and the Fisher's speciesproblem (given $n$ iid observations from an unknown distribution, the optimalprediction error of the number of unseen symbols in the next (unobserved) $r\cdot n$ observations is $\tilde\Theta(n^{-\min\{\frac{1}{r+1},\frac{1}{2}\}})$).

Dualizing Le Cam's method for functional estimation, with applications to estimating the unseens