convex body

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In mathematics, a convex body in $n$ -dimensional Euclidean space $\R^n$ 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 $\R^n.$

Some commonly known 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

$d(K,L) := \inf\{\epsilon \geq 0 : K \subset L + B^n(\epsilon), L \subset K + B^n(\epsilon) \}$ .{{Cite journal |last1=Hug |first1=Daniel |last2=Weil |first2=Wolfgang |date=2020 |title=Lectures on Convex Geometry |url=http://dx.doi.org/10.1007/978-3-030-50180-8 |journal=Graduate Texts in Mathematics |volume=286 |doi=10.1007/978-3-030-50180-8 |isbn=978-3-030-50179-2 |issn=0072-5285|url-access=subscription }}

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.

References

{{cite book |last1=Hiriart-Urruty |first1=Jean-Baptiste |last2=Lemaréchal |first2=Claude |title=Fundamentals of Convex Analysis |date=2001 |doi=10.1007/978-3-642-56468-0 |isbn=978-3-540-42205-1 |url=https://link.springer.com/book/10.1007/978-3-642-56468-0 |language=en}}
{{cite book |last1=Rockafellar |first1=R. Tyrrell |title=Convex Analysis |date=12 January 1997 |publisher=Princeton University Press |isbn=978-0-691-01586-6 |url=https://books.google.com/books?id=1TiOka9bx3sC&dq=Convex+Analysis%2C+Princeton+Mathematical+Series%2C+vol.+28&pg=PR7 |language=en}}
{{cite journal |last1=Arya |first1=Sunil |last2=Mount |first2=David M. |title=Optimal Volume-Sensitive Bounds for Polytope Approximation |journal=39th International Symposium on Computational Geometry (SoCG 2023) |date=2023 |volume=258 |pages=9:1–9:16 |doi=10.4230/LIPIcs.SoCG.2023.9|doi-access=free }}
{{cite journal| last=Gardner | first=Richard J. | title=The Brunn-Minkowski inequality | journal=Bull. Amer. Math. Soc. (N.S.) | volume=39 | issue=3 | year=2002 | pages=355–405 (electronic) | doi=10.1090/S0273-0979-02-00941-2 | doi-access=free }}

Category:Convex geometry

Category:Multi-dimensional geometry

convex body

Metric space structure

Polar body

See also

References