Complete and Reduced Convex Bodies

  • Horst MartiniEmail author
  • Luis Montejano
  • Déborah Oliveros


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.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Horst Martini
    • 1
    Email author
  • Luis Montejano
    • 2
  • Déborah Oliveros
    • 2
  1. 1.Faculty of MathematicsChemnitz University of TechnologyChemnitzGermany
  2. 2.Instituto de MatemáticasUniversidad Nacional Autónoma de México, Campus JuriquillaQuerétaroMéxico

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