# Convex body

In mathematics, a **convex body** in *n*-dimensional Euclidean space 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. 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 **R**^{n}.

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

## See also

## 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.

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