Brunn–Minkowski theorem

For measurable sets in Euclidean space, the volume of their Minkowski sum is at least the corresponding sum of their volume roots, raised to the ambient dimension.

Connect

Minkowski addition: The set of all sums formed by adding one point from each of two sets. The theorem bounds the volume of precisely this operation.
Euclidean space: A space modeled on finite-dimensional real coordinate vectors with the usual distance and inner product. The theorem concerns measurable sets inside this ambient setting.
Hermann Brunn: A German mathematician whose work on convex bodies contributed to the Brunn–Minkowski theory. Brunn established an early form of the volume inequality.
Isoperimetric inequality: An inequality relating a region's volume to the size of its boundary. Brunn–Minkowski provides a route to classical bounds on boundary area and volume.
Brunn's theorem: A theorem stating concavity of the root of cross-sectional volume for suitable convex bodies. It is a related concavity result that concerns sections rather than sums of sets.
Lebesgue measure: A measure that assigns lengths, areas, and volumes to broad classes of subsets of Euclidean space. It supplies the volume notion used in the measurable-set formulation.
Measurable set: A set to which a specified measure can consistently assign a size. Measurability ensures the sets in the theorem have well-defined volumes.
Hermann Minkowski: A German mathematician who developed geometry of numbers and foundational ideas about convex bodies. Minkowski's work gave the theorem its modern geometric framework.
Prékopa–Leindler inequality: An integral inequality that implies log-concavity properties and generalizes Brunn–Minkowski. It transfers the theorem's volume principle from sets to functions.
Minkowski's first inequality: An inequality relating mixed volume to the volumes of two convex bodies. It is a closely related convex-body inequality with a different volume comparison.