# Convex body

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A dodecahedron is a 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.

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 ${\displaystyle \mathbb {R} ^{n}.}$

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

## 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.