Convex body

In mathematics, a convex body in

-dimensional Euclidean space

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

is called symmetric if it is centrally symmetric with respect to the origin; that is to say, a point

if and only if its antipode,

Symmetric convex bodies are in a one-to-one correspondence with the unit balls of norms on

Some commonly known examples of convex bodies are the Euclidean ball, the hypercube and the cross-polytope.

for the set of convex bodies in

is a complete metric space with metric

) := inf { ϵ ≥ 0 :

[1] Further, the Blaschke Selection Theorem says that every d-bounded sequence in

has a convergent subsequence.

is a bounded convex body containing the origin

in its interior, the polar body

The polar body has several nice properties including

The polar body is a type of duality relation.