Convex body

A convex body in $\mathbb {R} ^{n}$ is a compact convex set with non-empty interior.

A convex body $K$ is called symmetric if it is centrally symmetric with respect to the origin, i.e. if $x\in K\iff x\in K$. Symmetric convex bodies are in a one-to-one correspondance with the unit balls of norms on $\mathbb {R} ^{n}$.

Important examples of convex bodies are the Euclidean ball, the hypercube and the cross-polytope.