Convex body

A dodecahedron is a 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 ${𝒦}^n$ for the set of convex bodies in ${ℝ}^n$. Then ${𝒦}^n$ is a complete metric space with metric $$d(K,L):=inf\{ϵ\ge 0:K\subset L+B^n(ϵ),L\subset K+B^n(ϵ)\}.$$

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

Polar body

If $K$ is a bounded convex body containing the origin $O$ in its interior, the polar body $K^{\ast }$ is $\{u:⟨u,v⟩\le 1,\mathrm{\forall }v\in K\}$. The polar body has several nice properties including $(K^{\ast }{)}^{\ast }=K$, $K^{\ast }$ is bounded, and if $K_1\subset K_2$ then $K_2^{\ast }\subset K_1^{\ast }$. 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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