In mathematics, a convex body in -dimensional Euclidean space 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.
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A convex body is called symmetric if it is centrally symmetric with respect to the origin; that is to say, a point lies in if and only if its antipode, also lies in Symmetric convex bodies are in a one-to-one correspondence with the unit balls of norms on
Important examples of convex bodies are the Euclidean ball, the hypercube and the cross-polytope.
Metric space structure
editWrite for the set of convex bodies in . Then is a complete metric space with metric
.[1]
Further, the Blaschke Selection Theorem says that every d-bounded sequence in has a convergent subsequence.[1]
Polar body
editIf is a bounded convex body containing the origin in its interior, the polar body is . The polar body has several nice properties including , is bounded, and if then . The polar body is a type of duality relation.
See also
edit- List of convexity topics
- John ellipsoid
- Brunn–Minkowski theorem, which has many implications relevant to the geometry of convex bodies.
References
edit- ^ a b 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.