  Lower bounds involving $f$-divergences between the underlying probability
measures are proved for the minimax risk in estimation problems. Our proofs
just use simple convexity facts. Special cases and straightforward corollaries
of our bounds include well known inequalities for establishing minimax lower
bounds such as Fano's inequality, Pinsker's inequality and inequalities based
on global entropy conditions. Two applications are provided: a new minimax
lower bound for the reconstruction of convex bodies from noisy support function
measurements and a different proof of a recent minimax lower bound for the
estimation of a covariance matrix.
