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. 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 ${\displaystyle K}$ 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.

Metric space structure

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

${\displaystyle d(K,L):=\inf\{\epsilon \geq 0:K\subset L+B^{n}(\epsilon ),L\subset K+B^{n}(\epsilon )\}}$.^[1]

Further, the Blaschke Selection Theorem says that every d-bounded sequence in ${\displaystyle {\mathcal {K}}^{n}}$ has a convergent subsequence.^[1]

Polar body

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

See also

• List of convexity topics
• John ellipsoid
• Brunn–Minkowski theorem, which has many implications relevant to the geometry of convex bodies.

References

1. Hug, Daniel; Weil, Wolfgang (2020). "Lectures on Convex Geometry". Graduate Texts in Mathematics. 286. doi:10.1007/978-3-030-50180-8. ISBN 978-3-030-50179-2. ISSN 0072-5285.

• 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.