Convex body

In mathematics, a convex body in $n$-dimensional Euclidean space $\mathbb {R} ^{n}$ is a compact convex set with non-empty interior. Some authors do not require a non-empty interior, merely that the set is non-empty.

A dodecahedron is a convex body.

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.

Metric space structure

Write ${\mathcal {K}}^{n}$ for the set of convex bodies in $\mathbb {R} ^{n}$. Then ${\mathcal {K}}^{n}$ is a complete metric space with metric

$d(K,L):=\inf\{\epsilon \geq 0:K\subset L+B^{n}(\epsilon ),L\subset K+B^{n}(\epsilon )\}$.

Further, the Blaschke Selection Theorem says that every d-bounded sequence in ${\mathcal {K}}^{n}$ has a convergent subsequence.

Polar body

If $K$ is a bounded convex body containing the origin $O$ in its interior, the polar body $K^{*}$ is $\{u:\langle u,v\rangle \leq 1,\forall v\in K\}$. The polar body has several nice properties including $(K^{*})^{*}=K$, $K^{*}$ is bounded, and if $K_{1}\subset K_{2}$ then $K_{2}^{*}\subset K_{1}^{*}$. The polar body is a type of duality relation.

See also

• John ellipsoid
• List of convexity topics

References

• Hiriart-Urruty, Jean-Baptiste; Lemaréchal, Claude (2001). Fundamentals of Convex Analysis. doi:10.1007/978-3-642-56468-0. ISBN 978-3-540-42205-1.
• Rockafellar, R. Tyrrell (12 January 1997). Convex Analysis. Princeton University Press. ISBN 978-0-691-01586-6.
• Arya, Sunil; Mount, David M. (2023). "Optimal Volume-Sensitive Bounds for Polytope Approximation". 39th International Symposium on Computational Geometry (SoCG 2023). 258: 9:1–9:16. doi:10.4230/LIPIcs.SoCG.2023.9.
• 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.