Linear hyperspectral unmixing (HU) aims at factoring the observation matrix into an endmember matrix and an abundance matrix . Linear HU via variational minimum volume (MV) regularization has recently received considerable attention in the remote sensing and machine learning areas, mainly owing to its robustness against the absence of pure pixels. We put some popular linear HU formulations under a unifying framework, which involves a data-fitting term and an MV-based regularization term, and collectively solve it via a nonconvex optimization. As the former and the latter terms tend, respectively, to expand (reducing the data-fitting errors) and to shrink the simplex enclosing the measured spectra, it is critical to strike a balance between those two terms. To the best of our knowledge, the existing methods find such balance by tuning a regularization parameter manually , which has little value in unsupervised scenarios. In this paper, we aim at selecting the regularization parameter automatically by exploiting the fact that a too large parameter overshrinks the volume of the simplex defined by the endmembers, making many data points be left outside of the simplex and hence inducing a large data-fitting error, while a sufficiently small parameter yields a large simplex making data-fitting error very small. Roughly speaking, the transition point happens when the simplex still encloses the data cloud but there are data points on all its facets. These observations are systematically formulated to find the transition point that, in turn, yields a good parameter. The competitiveness of the proposed selection criterion is illustrated with simulated and real data.