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 $ε^{*2} \wedge \operatorname{diam}(K)^2$ with \begin{align*} ε^* = \sup \bigg\{ε: \frac{ε^2}{σ^2} \leq \log M^{\operatorname{loc}}(ε)\bigg\}, \end{align*} where $\log M^{\operatorname{loc}}(ε)$ denotes the local entropy of the set $K$, and $σ^2$ 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 $σ^2$ to show that the minimax rate in that case is $ε^{*2}$.