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 $$\R^n.$$

Important 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

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

Further, the Blaschke Selection Theorem says that every d-bounded sequence in $$\mathcal K^n$$ 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 $$\{u : \langle u,v \rangle \leq 1, \forall v \in K \} $$. 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.