Convex body

In mathematics, a convex body in n-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ is a compact convex set with non-empty interior.

A dodecahedron is a convex body.

A convex body K is called symmetric if it is centrally symmetric with respect to the origin, i.e. 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 Rⁿ.

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

See also

• List of convexity topics

References

• Gardner, Richard J. (2002). "The Brunn-Minkowski inequality". Bull. Amer. Math. Soc. (N.S.). 39 (3): 355–405 (electronic). doi:10.1090/S0273-0979-02-00941-2.