Convex body

In mathematics, a convex body in ${\displaystyle n}$-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ 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 dodecahedron is a convex body.

A convex body ${\displaystyle K}$ is called symmetric if it is centrally symmetric with respect to the origin; that is to say, a point ${\displaystyle x}$ lies in ${\displaystyle K}$ if and only if its antipode, ${\displaystyle -x}$ also lies in ${\displaystyle K.}$ Symmetric convex bodies are in a one-to-one correspondence with the unit balls of norms on ${\displaystyle \mathbb {R} ^{n}.}$

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

Metric space structure

Write ${\displaystyle {\mathcal {K}}^{n}}$ for the set of convex bodies in ${\displaystyle \mathbb {R} ^{n}}$. Then ${\displaystyle {\mathcal {K}}^{n}}$ is a complete metric space with metric

${\displaystyle d(K,L):=\inf\{\epsilon \geq 0:K\subset L+B^{n}(\epsilon ),L\subset K+B^{n}(\epsilon )\}}$.^[1]

Further, the Blaschke Selection Theorem says that every d-bounded sequence in ${\displaystyle {\mathcal {K}}^{n}}$ has a convergent subsequence.^[1]

Polar body

If ${\displaystyle K}$ is a bounded convex body containing the origin ${\displaystyle O}$ in its interior, the polar body ${\displaystyle K^{*}}$ is ${\displaystyle \{u:\langle u,v\rangle \leq 1,\forall v\in K\}}$. The polar body has several nice properties including ${\displaystyle (K^{*})^{*}=K}$, ${\displaystyle K^{*}}$ is bounded, and if ${\displaystyle K_{1}\subset K_{2}}$ then ${\displaystyle K_{2}^{*}\subset K_{1}^{*}}$. The polar body is a type of duality relation.

See also

• List of convexity topics
• John ellipsoid – Ellipsoid most closely containing, or contained in, an n-dimensional convex object
• Brunn–Minkowski theorem, which has many implications relevant to the geometry of convex bodies.