Convex and affine geometry in n-dimensions provide powerful tools for the analysis, understanding, and visualization of hyperspectral data. The ubiquitous mixed pixel problem can be exploited as an advantage and is easily cast in an n-d convexity context. Convexity concepts can be used to identify the purest pixels in a given scene and to unravel spectral mixing, both fully and partially. Visualization techniques based on these concepts permit human interpretation of all spectral information of all image pixels simultaneously. Convex geometry forms a natural framework for the unique challenges associated with analysis of hyperspectral data.