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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 $$L_2$$ 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 $$\epsilon^{*2}$$.