Convex body explained
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
Some commonly known examples of convex bodies are the Euclidean ball, the hypercube and the cross-polytope.
Metric space structure
Write
for the set of convex bodies in
. Then
is a
complete metric space with metric
d(K,L):=inf\{\epsilon\geq0:K\subsetL+Bn(\epsilon),L\subsetK+Bn(\epsilon)\}
.
[1] Further, the Blaschke Selection Theorem says that every d-bounded sequence in
has a convergent subsequence.
Polar body
If
is a bounded convex body containing the origin
in its interior, the
polar body
is
\{u:\langleu,v\rangle\leq1,\forallv\inK\}
. The polar body has several nice properties including
,
is bounded, and if
then
. The polar body is a type of
duality relation.
See also
References
- Book: Hiriart-Urruty . Jean-Baptiste . Lemaréchal . Claude . Fundamentals of Convex Analysis . 2001 . 10.1007/978-3-642-56468-0 . 978-3-540-42205-1 . en.
- Book: Rockafellar . R. Tyrrell . Convex Analysis . 12 January 1997 . Princeton University Press . 978-0-691-01586-6 . en.
- Arya . Sunil . Mount . David M. . Optimal Volume-Sensitive Bounds for Polytope Approximation . 39th International Symposium on Computational Geometry (SoCG 2023) . 2023 . 258 . 9:1–9:16 . 10.4230/LIPIcs.SoCG.2023.9. free .
- Gardner . Richard J. . The Brunn-Minkowski inequality . Bull. Amer. Math. Soc. (N.S.) . 39 . 3 . 2002 . 355 - 405 (electronic) . 10.1090/S0273-0979-02-00941-2 . free .
Notes and References
- Hug . Daniel . Weil . Wolfgang . 2020 . Lectures on Convex Geometry . Graduate Texts in Mathematics . 286 . 10.1007/978-3-030-50180-8 . 978-3-030-50179-2 . 0072-5285.
