In mathematics, a convex body in n-dimensional Euclidean space \R^n is a compact convex set with non-empty interior. Some authors do not require a non-empty interior, merely that the set is non-empty. A convex body K is called symmetric if it is centrally symmetric with respect to the origin; that is to say, a point x lies in K if and only if its antipode, - x also lies in K. Symmetric convex bodies are in a one-to-one correspondence with the unit balls of norms on \R^n. Some commonly known examples of convex bodies are the Euclidean ball, the hypercube and the cross-polytope.

Metric space structure

Write for the set of convex bodies in \mathbb R^n. Then is a complete metric space with metric . Further, the Blaschke Selection Theorem says that every d-bounded sequence in has a convergent subsequence.

Polar body

If K is a bounded convex body containing the origin O in its interior, the polar body K^* is. The polar body has several nice properties including (K^)^=K, K^* is bounded, and if then. The polar body is a type of duality relation.