  We consider a convex constrained Gaussian sequence model and characterize
necessary and sufficient conditions for the least squares estimator (LSE) to be
optimal in a minimax sense. For a closed convex set $K\subset \mathbb{R}^n$ we
observe $Y=\mu+\xi$ for $\xi\sim N(0,\sigma^2\mathbb{I}_n)$ and $\mu\in K$ and
aim to estimate $\mu$. We characterize the worst case risk of the LSE in
multiple ways by analyzing the behavior of the local Gaussian width on $K$. 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 $\ell_p$ balls for
$p\in[1,2]$, pyramids, solids of revolution, and multivariate isotonic
regression, among others.
