We consider a convex constrained Gaussian sequence model and characterize necessary and sufficient conditions for the least squares estimator (LSE) to be minimax optimal. For a closed convex setK⊂ Rnwe observeY= μ + ξ for ξ ~N(0,σ2In) and μ ∈Kand aim to estimate μ. We characterize the worst case risk of the LSE in multiple ways by analyzing the behavior of the local Gaussian width onK. We demonstrate that optimality is equivalent to a Lipschitz property of the local Gaussian width mapping. We also provide theoretical algorithms that search for the worst case risk. We then provide examples showing optimality or suboptimality of the LSE on various sets, including ℓpballs forp∈ [1,2], pyramids, solids of revolution, and multivariate isotonic regression, among others.