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In a standard linear model, we explore the optimality of the least squares estimator under assuptions stronger than those for the Gauss-Markov theorem. The conclusion is then much stronger than that of the Gauss-Markov theorem. Specifically, two results are cited below: Under the assumption that the unobserved error e has a spherically symmetric distribution, the least squares estimator for the regression coefficient β is shown to maximize the probability that β − β stays in any symmetric convex set among linear unbiased estimators β. With the additional assumption that e is unimodal, the conclusion holds among equivariant estimators. The import of these results for risk functions is also discussed.