If $\rho_{12} \lt 1$ or $\sigma_1 \ne \sigma_2$ then $\sigma_{V}^2$ representing the variance of the portfolio with weights $(w_1, w_2)=(s, 1-s)$ as a function of $s$ attains its minimum value at
$$s_0=\frac{\sigma_{2}^{2}-\sigma_1 \sigma_2 \rho_{12}}{\sigma_{1}^{2} + \sigma_{2}^{2} - 2\sigma_1 \sigma_2 \rho_{12}}$$
Under which conditions on $\sigma_1, \sigma_2$ and $\rho_{12}$ the minimum variance portfolio involves no short selling?
If I understand the question correctly the portfolio attains its minimum at $s_0$ so if I want to find constraints for $\sigma_1, \sigma_2$ and $\rho_{12}$ that involves no short selling I have to make sure that $s_0 \gt 0$?