Counterfeit pharmaceutical products pose a serious public health problem. It is thus important not only to detect them, but also to identify their composition and assess the risk for the patient. Identifying the spectral signatures of the pure compounds present in a (maybe counterfeit) tablet of unknown origin is clearly a hyperspectral unmixing problem. In fact, under a linear mixing model, the hyperspectral vectors belong to a simplex whose vertices are the spectra of the pure compounds in the mixture. Minimum volume simplex analysis (MVSA) and minimum-volume enclosing simplex (MVES) are recently proposed algorithms, exploiting the idea of finding a simplex of minimum volume fitting the observed data. This work gives evidence of the usefulness of MVES and MVSA for unmixing near infrared (NIR) hyperspectral data of tablets of unknown composition. Experiments reported in this paper show that MVES and MVSA strongly outperform the state-of-the-art method in analytical chemistry for spectral unmixing: multivariate curve resolution - alternating least squares (MCR-ALS). These experiments are based on synthetic data (studying the effect of noise and of the presence/absence of pure pixels) and on a real dataset composed of NIR hyperspectral images of counterfeit tablets.