When can the minimum volume enclosing simplex identify the endmembers correctly when there is no pure pixel?

In blind hyperspectral unmixing, it has been commonly believed that the minimum volume enclosing simplex (MVES) criterion is robust against lack of pure pixels. Specifically, such a belief has been based on empirical experience, where extensive numerical results showed that MVES-based algorithms may identify the underlying endmembers quite accurately under high signal-to-noise ratios and without pure pixels. In this paper, we report some theoretical results on the endmember identifiability of the MVES criterion in the noiseless case. We employ an assumption that is a two-mixture generalization of the pure-pixel assumption; particularly, we require a set of pixels, each being constituted by only two endmembers (rather than one as in the pure-pixel assumption), to exist in the data set. Under this assumption and some rather mild condition, we show that the MVES solution perfectly identifies the true endmembers. Numerical simulation results are provided to verify our theoretical results.