Complete and Reduced Convex Bodies

Abstract

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 compact 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 Pal, 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.