Convex body

Short description: Non-empty convex set in Euclidean space

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. Some authors do not require a non-empty interior, merely that the set is non-empty.

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 ${\mathbb{ℝ}}^n.$

Some commonly known examples of convex bodies are the Euclidean ball, the hypercube and the cross-polytope.

Metric space structure

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

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

Further, the Blaschke Selection Theorem says that every d-bounded sequence in ${{𝒦}}^n$ 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 $\{u:⟨u,v⟩\le 1,\mathrm{\forall }v\in K\}$. 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

• List of convexity topics
• John ellipsoid – Ellipsoid most closely containing, or contained in, an n-dimensional convex object
• Brunn–Minkowski theorem, which has many implications relevant to the geometry of convex bodies.

References

• Hiriart-Urruty, Jean-Baptiste; Lemaréchal, Claude (2001) (in en). Fundamentals of Convex Analysis. doi:10.1007/978-3-642-56468-0. ISBN 978-3-540-42205-1. https://link.springer.com/book/10.1007/978-3-642-56468-0. 
• Rockafellar, R. Tyrrell (12 January 1997) (in en). Convex Analysis. Princeton University Press. ISBN 978-0-691-01586-6. https://books.google.com/books?id=1TiOka9bx3sC&dq=Convex+Analysis%2C+Princeton+Mathematical+Series%2C+vol.+28&pg=PR7. 
• Arya, Sunil; Mount, David M. (2023). "Optimal Volume-Sensitive Bounds for Polytope Approximation". 39th International Symposium on Computational Geometry (SoCG 2023) (Schloss Dagstuhl – Leibniz-Zentrum für Informatik) 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.