From Wikipedia, the free encyclopedia
Non-empty convex set in Euclidean space
A
dodecahedron is a convex body.
In
mathematics , a convex body in
n
{\displaystyle n}
-
dimensional
Euclidean space
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}
is called symmetric if it is centrally symmetric with respect to the origin; that is to say, a point
x
{\displaystyle x}
lies in
K
{\displaystyle K}
if and only if its
antipode ,
−
x
{\displaystyle -x}
also lies in
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}.}
Important examples of convex bodies are the
Euclidean ball , the
hypercube and the
cross-polytope .
Write
K
n
{\displaystyle {\mathcal {K}}^{n}}
for the set of convex bodies in
R
n
{\displaystyle \mathbb {R} ^{n}}
. Then
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 )\}}
.
[1]
Further, the
Blaschke Selection Theorem says that every d -bounded sequence in
K
n
{\displaystyle {\mathcal {K}}^{n}}
has a convergent subsequence.
[1]
If
K
{\displaystyle K}
is a bounded convex body containing the origin
O
{\displaystyle O}
in its interior, the polar body
K
∗
{\displaystyle K^{*}}
is
{
u
:
⟨
u
,
v
⟩
≤
1
,
∀
v
∈
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}
,
K
∗
{\displaystyle K^{*}}
is bounded, and if
K
1
⊂
K
2
{\displaystyle K_{1}\subset K_{2}}
then
K
2
∗
⊂
K
1
∗
{\displaystyle K_{2}^{*}\subset K_{1}^{*}}
. The polar body is a type of
duality relation.