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]

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.

• 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.

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