Inmathematics, a convex bodyin-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 ballsofnormson
Important examples of convex bodies are the Euclidean ball, the hypercube and the cross-polytope.
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]
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.
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Basic concepts |
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Topics (list) |
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Maps |
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Main results (list) |
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Sets |
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Series |
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Duality |
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Applications and related |
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