Common-subspace volume power means hit a sharp threshold: one factor is reachable in polynomial time, anything larger is NP-hard
For compressing several positive semidefinite matrices into one low-dimensional subspace and scoring the result by a weighted power mean of volume terms, the optimal approximation factor is now pinned down independently of ambient dimension: for every fixed subspace dimension and fixed rational power, a randomized polynomial-time algorithm attains it while any larger constant factor is NP-hard. Above a critical power the factor is an explicit product of Gamma functions; at and below it, every positive constant-factor approximation is NP-hard. The method is a determinant-preserving concave relaxation over trace-one positive semidefinite matrices followed by Gaussian rounding back to a single subspace.