The Convex Hull of Knowledge

The convex hull of a set of points is the smallest convex shape containing all of them — the rubber band stretched around a set of pins. Every point inside can be reached by weighted combinations of the pins; every point outside cannot. Applied to large language models, the training corpus defines a set of points in a very high-dimensional space of possible texts. The convex hull is the set of all texts reachable by combining patterns from the training data. The model's outputs, governed by the statistical regularities it has learned, lie within this hull or in its immediate neighborhood. A point outside the hull would require variation that the model's optimization explicitly penalizes. This is the geometric structure behind Campbell's claim that AI's variation is always directed — and the frame that reveals why genuine discovery requires something the technology's architecture structurally prevents.