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 $\mathbb {R} ^{n}.$

Important 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

$d(K,L):=\inf\{\epsilon \geq 0:K\subset L+B^{n}(\epsilon ),L\subset K+B^{n}(\epsilon )\}$ .^[1]

Further, the Blaschke Selection Theorem says that every d-bounded sequence in ${\mathcal {K}}^{n}$ has a convergent subsequence.^[1]

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

^ ^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.

[:0-1] 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.

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v t e Convex analysis and variational analysis
Basic concepts	Convex combination Convex function Convex set
Topics (list)	Choquet theory Convex geometry Convex metric space Convex optimization Duality Lagrange multiplier Legendre transformation Locally convex topological vector space Simplex
Maps	Convex conjugate Concave (Closed K- Logarithmically Proper Pseudo- Quasi-) Convex function Invex function Legendre transformation Semi-continuity Subderivative
Main results (list)	Carathéodory's theorem Ekeland's variational principle Fenchel–Moreau theorem Fenchel-Young inequality Jensen's inequality Hermite–Hadamard inequality Krein–Milman theorem Mazur's lemma Shapley–Folkman lemma Robinson–Ursescu Simons Ursescu
Sets	Convex hull (Orthogonally, Pseudo-) Convex set Effective domain Epigraph Hypograph John ellipsoid Lens Radial set/Algebraic interior Zonotope
Series	Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx))
Duality	Dual system Duality gap Strong duality Weak duality
Applications and related	Convexity in economics

Metric space structure

Polar body

See also

References