convex body

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

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

Metric space structure

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

$d(K,L) := \inf\$.

Further, the Blaschke Selection Theorem says that every ''d''-bounded sequence in $\mathcal K^n$ has a convergent subsequence.

Polar body

If $K$ is a bounded convex body containing the origin $O$ in its interior, the polar body $K^*$ is $\$. The polar body has several nice properties including $(K^*)^*=K$, $K^*$ is bounded, and if $K_1\subset K_2$ then $K_2^*\subset K_1^*$. The polar body is a type of duality relation.

See also

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* Brunn–Minkowski theorem, which has many implications relevant to the geometry of convex bodies.

References

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Convex geometry
Multi-dimensional geometry