Convex body

A dodecahedron is a convex body.

In mathematics, a convex body in $n$-dimensional Euclidean space $\mathbb {R} ^{n}$ 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; that is to say, 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 $\mathbb {R} ^{n}.$

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

See also

• List of convexity topics
• John ellipsoid – Ellipsoid most closely containing, or contained in, an n-dimensional convex object

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.