A new geometrically-motivated algorithm for topic modeling is developed and applied to the discovery of latent “topics” in text and image “document” corpora. The algorithm is based on robustly finding and clustering extreme-points of empirical cross-document word-frequencies that correspond to novel words unique to each topic. In contrast to related approaches that are based on solving non-convex optimization problems using suboptimal approximations, locally-optimal methods, or heuristics, the new algorithm is convex, has polynomial complexity, and has competitive qualitative and quantitative performance compared to the current state- of-the-art approaches on synthetic and real-world datasets.