In this paper, we derive simple, directly computable conditions for minimum-variance portfolios to have all positive weights. We show that either there is no minimum-variance portfolio with all positive weights or there is a single segment of the minimum-variance frontier for which all portfolios have positive weights. Then, we examine the likelihood of observing positively weighted minimum-variance portfolios. Analytical and computational results suggest that: i) even if the mean vector and covariance matrix are compatible with a given positively weighted portfolio being mean-variance efficient, the proportion of the minimum-variance frontier containing positively weighted portfolios is small and decreases as the number of assets in the universe increases, and ii) small perturbations in the means will likely lead to no positively weighted minimum-variance portfolios.