  We study the classical problem of deriving minimax rates for density
estimation over convex density classes. Building on the pioneering work of Le
Cam (1973), Birge (1983, 1986), Wong and Shen (1995), Yang and Barron (1999),
we determine the exact (up to constants) minimax rate over any convex density
class. This work thus extends these known results by demonstrating that the
local metric entropy of the density class always captures the minimax optimal
rates under such settings. Our bounds provide a unifying perspective across
both parametric and nonparametric convex density classes, under weaker
assumptions on the richness of the density class than previously considered.
Our proposed `multistage sieve' MLE applies to any such convex density class.
We further demonstrate that this estimator is also adaptive to the true
underlying density of interest. We apply our risk bounds to rederive known
minimax rates including bounded total variation, and Holder density classes. We
further illustrate the utility of the result by deriving upper bounds for less
studied classes, e.g., convex mixture of densities.
