The Brunn-Minkowski theorem

The Brunn-Minkowski theorem is a key result in geometry that relates the volume of two shapes to the volume of their combined shape when they are merged. It states that, for two sets in space, the measure of their Minkowski sum (adding every point of one to every point of the other) grows in a way that’s at least as fast as a certain root of their individual measures. In essence, combining shapes results in a volume that isn't less than a particular mathematical blend of the original volumes, emphasizing the idea that combined shapes tend to be larger in a predictable way.