Convex body

In mathematics, a convex body in ${\displaystyle n}$-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ 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 ${\displaystyle K}$ is called symmetric if it is centrally symmetric with respect to the origin; that is to say, a point ${\displaystyle x}$ lies in ${\displaystyle K}$ if and only if its antipode, ${\displaystyle -x}$ also lies in ${\displaystyle K.}$ Symmetric convex bodies are in a one-to-one correspondence with the unit balls of norms on ${\displaystyle \mathbb {R} ^{n}.}$

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