The analysis of hyperspectral data sets requires the determination of certain basis spectra called 'end-members.' Once these spectra are found, the image cube can be 'unmixed' into the fractional abundance of each material in each pixel. There exist several techniques for accomplishing the determination of the end-members, most of which involve the intervention of a trained geologist. Often these-end-members are assumed to be present in the image, in the form of pure, or unmixed, pixels. In this paper a method based upon the geometry of convex sets is proposed to find a unique set of purest pixels in an image. The technique is based on the fact that in N spectral dimensions, the N-volume contained by a simplex formed of the purest pixels is larger than any other volume formed from any other combination of pixels. The algorithm works by 'inflating' a simplex inside the data, beginning with a random set of pixels. For each pixel and each end-member, the end-member is replaced with the spectrum of the pixel and the volume is recalculated. If it increases, the spectrum of the new pixel replaces that end-member. This procedure is repeated until no more replacements are done. This algorithm successfully derives end-members in a synthetic data set, and appears robust with less than perfect data. Spectral end-members have been extracted for the AVIRIS Cuprite data set which closely match reference spectra, and resulting abundance maps match published mineral maps.