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.

Kinds of convex bodies

A convex body may be defined as:

1. A Convex set of points.
2. The Convex Hull of a set of points.
3. The intersection of Hyperplanes.
4. The interior of any Convex polygon or Convex polytope.

Polar body

If $K$ is a bounded convex body containing the origin $O$ in it's 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 – Ellipsoid most closely containing, or contained in, an n-dimensional convex object
• List of convexity topics

References

• 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.{{cite journal}}: CS1 maint: unflagged free DOI (link)
• 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.