Convex body

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A dodecahedron is a convex body.

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.

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[edit]

Write 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[edit]

If 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]

References[edit]

  1. ^ a b Hug, Daniel; Weil, Wolfgang (2020). "Lectures on Convex Geometry". Graduate Texts in Mathematics. doi:10.1007/978-3-030-50180-8. ISSN 0072-5285.