The Minkowski-Weyl theorem

The Minkowski-Weyl theorem states that any shape in space that is both convex (no indentations) and bounded (finite size) can be described in two equivalent ways: as a combination of flat faces forming a polyhedron, or as the intersection of a finite number of half-spaces (regions divided by flat boundaries). It shows that complex convex shapes can be understood either by their outward surfaces or by simpler boundary conditions, linking their geometric structure to algebraic descriptions. This fundamental result bridges geometric intuition and algebraic methods in convex analysis.