  We study the optimality of the minimax risk of truncated series estimators
for symmetric convex polytopes. We show that the optimal truncated series
estimator is within $O(\log m)$ factor of the optimal if the polytope is
defined by $m$ hyperplanes. This represents the first such bounds towards
general convex bodies. In proving our result, we first define a geometric
quantity, called the \emph{approximation radius}, for lower bounding the
minimax risk. We then derive our bounds by establishing a connection between
the approximation radius and the Kolmogorov width, the quantity that provides
upper bounds for the truncated series estimator. Besides, our proof contains
several ingredients which might be of independent interest: 1. The notion of
approximation radius depends on the volume of the body. It is an intuitive
notion and is flexible to yield strong minimax lower bounds; 2. The connection
between the approximation radius and the Kolmogorov width is a consequence of a
novel duality relationship on the Kolmogorov width, developed by utilizing some
deep results from convex geometry.
