  We determine the exact minimax rate of a Gaussian sequence model under
bounded convex constraints, purely in terms of the local geometry of the given
constraint set $K$. Our main result shows that the minimax risk (up to constant
factors) under the squared $\ell_2$ loss is given by $\epsilon^{*2} \wedge
\operatorname{diam}(K)^2$ with \begin{align*}
  \epsilon^* = \sup \bigg\{\epsilon : \frac{\epsilon^2}{\sigma^2} \leq \log
M^{\operatorname{loc}}(\epsilon)\bigg\}, \end{align*} where $\log
M^{\operatorname{loc}}(\epsilon)$ denotes the local entropy of the set $K$, and
$\sigma^2$ is the variance of the noise. We utilize our abstract result to
re-derive known minimax rates for some special sets $K$ such as
hyperrectangles, ellipses, and more generally quadratically convex
orthosymmetric sets. Finally, we extend our results to the unbounded case with
known $\sigma^2$ to show that the minimax rate in that case is $\epsilon^{*2}$.
