Complete and Reduced Convex Bodies

We say that a compact set in $$\mathbb {E}^n$$ is complete (or diametrically complete) if, adding any point to it, its diameter will increase. If we take the partially ordered set $$\Omega _h$$ of all compactBody complete sets of diameter h in n-dimensional Euclidean space ordered by inclusion, complete bodies are precisely the maximal elements of $$\Omega _h$$ . That is, a compact set A in $$\Omega _h$$ is a maximal element of $$\Omega _h$$ , or a complete body, if A is equal to B whenever A is contained in B, for B in $$\Omega _h$$ . The two main results of this chapter are that complete bodies are precisely bodies of constant width h, and that every element of $$\Omega _h$$ is contained in a maximal body; that is, that it can be completed to a body of constant width. These results are known as the Theorems of Meissner and Pál, respectively. Section 7.4 will be devoted to the study of reduced convex bodies, a notion somehow “dual” to completeness, and in Section 7.5 we complete convex bodies preserving some of their original characteristics, such as symmetries.