Convex body (original) (raw)

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Non-empty convex set in Euclidean space

A dodecahedron is a convex body.

In mathematics, a convex body in n {\displaystyle n} {\displaystyle n}-dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} {\displaystyle \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 {\displaystyle K} {\displaystyle K} is called symmetric if it is centrally symmetric with respect to the origin; that is to say, a point x {\displaystyle x} {\displaystyle x} lies in K {\displaystyle K} {\displaystyle K} if and only if its antipode, − x {\displaystyle -x} {\displaystyle -x} also lies in K . {\displaystyle K.} {\displaystyle K.} Symmetric convex bodies are in a one-to-one correspondence with the unit balls of norms on R n . {\displaystyle \mathbb {R} ^{n}.} {\displaystyle \mathbb {R} ^{n}.}

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

Metric space structure

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

d ( K , L ) := inf { ϵ ≥ 0 : K ⊂ L + B n ( ϵ ) , L ⊂ K + B n ( ϵ ) } {\displaystyle d(K,L):=\inf\{\epsilon \geq 0:K\subset L+B^{n}(\epsilon ),L\subset K+B^{n}(\epsilon )\}} {\displaystyle 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 K n {\displaystyle {\mathcal {K}}^{n}} {\displaystyle {\mathcal {K}}^{n}} has a convergent subsequence.[1]

If K {\displaystyle K} {\displaystyle K} is a bounded convex body containing the origin O {\displaystyle O} {\displaystyle O} in its interior, the polar body K ∗ {\displaystyle K^{*}} {\displaystyle K^{*}} is { u : ⟨ u , v ⟩ ≤ 1 , ∀ v ∈ K } {\displaystyle \{u:\langle u,v\rangle \leq 1,\forall v\in K\}} {\displaystyle \{u:\langle u,v\rangle \leq 1,\forall v\in K\}}. The polar body has several nice properties including ( K ∗ ) ∗ = K {\displaystyle (K^{*})^{*}=K} {\displaystyle (K^{*})^{*}=K}, K ∗ {\displaystyle K^{*}} {\displaystyle K^{*}} is bounded, and if K 1 ⊂ K 2 {\displaystyle K_{1}\subset K_{2}} {\displaystyle K_{1}\subset K_{2}} then K 2 ∗ ⊂ K 1 ∗ {\displaystyle K_{2}^{*}\subset K_{1}^{*}} {\displaystyle K_{2}^{*}\subset K_{1}^{*}}. The polar body is a type of duality relation.

  1. ^ 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.