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Erschienen in:
2019 | OriginalPaper | Buchkapitel
7. Complete and Reduced Convex Bodies
verfasst von : Horst Martini, Luis Montejano, Déborah Oliveros
Erschienen in: Bodies of Constant Width
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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 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.