The Shapley–Folkman Lemma

The Shapley–Folkman Lemma states that when combining many small, convex sets (like groups of points or options), the overall sum is nearly convex, even if some individual sets are not perfectly convex. Specifically, as the number of sets increases, the “non-convexity” (irregularities) of their sum diminishes. This insight explains why large, complex systems often behave smoothly and predictably, despite having components with irregular shapes or properties. It’s a fundamental idea in areas like optimization and economics, showing that aggregating many small parts can lead to more stable, convex-like behavior.