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}.}

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

Metric space structure

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 )\}.}

Further, the Blaschke Selection Theorem says that every d-bounded sequence in

K

n

{\displaystyle {\mathcal {K}}^{n}}

has a convergent subsequence.

Polar body

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}^{*}}

.