Convex body
In mathematics, a convex body in [math]\displaystyle{ n }[/math]-dimensional Euclidean space [math]\displaystyle{ \R^n }[/math] 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 [math]\displaystyle{ K }[/math] is called symmetric if it is centrally symmetric with respect to the origin; that is to say, a point [math]\displaystyle{ x }[/math] lies in [math]\displaystyle{ K }[/math] if and only if its antipode, [math]\displaystyle{ - x }[/math] also lies in [math]\displaystyle{ K. }[/math] Symmetric convex bodies are in a one-to-one correspondence with the unit balls of norms on [math]\displaystyle{ \R^n. }[/math]
Important examples of convex bodies are the Euclidean ball, the hypercube and the cross-polytope.
Metric space structure
Write [math]\displaystyle{ \mathcal K^n }[/math] for the set of convex bodies in [math]\displaystyle{ \mathbb R^n }[/math]. Then [math]\displaystyle{ \mathcal K^n }[/math] is a complete metric space with metric
[math]\displaystyle{ d(K,L) := \inf\{\epsilon \geq 0 : K \subset L + B^n(\epsilon), L \subset K + B^n(\epsilon) \} }[/math].[1]
Further, the Blaschke Selection Theorem says that every d-bounded sequence in [math]\displaystyle{ \mathcal K^n }[/math] has a convergent subsequence.[1]
Polar body
If [math]\displaystyle{ K }[/math] is a bounded convex body containing the origin [math]\displaystyle{ O }[/math] in its interior, the polar body [math]\displaystyle{ K^* }[/math] is [math]\displaystyle{ \{u : \langle u,v \rangle \leq 1, \forall v \in K \} }[/math]. The polar body has several nice properties including [math]\displaystyle{ (K^*)^*=K }[/math], [math]\displaystyle{ K^* }[/math] is bounded, and if [math]\displaystyle{ K_1\subset K_2 }[/math] then [math]\displaystyle{ K_2^*\subset K_1^* }[/math]. The polar body is a type of duality relation.
See also
- John ellipsoid
- List of convexity topics – Wikipedia list article
References
- ↑ 1.0 1.1 Hug, Daniel; Weil, Wolfgang (2020). "Lectures on Convex Geometry". Graduate Texts in Mathematics. doi:10.1007/978-3-030-50180-8. ISSN 0072-5285. http://dx.doi.org/10.1007/978-3-030-50180-8.
- 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) 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.
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