On multi-way metricity, minimality and diagonal planes

  • Matthijs J. Warrens
Open Access
Regular Article


Validity of the triangle inequality and minimality, both axioms for two-way dissimilarities, ensures that a two-way dissimilarity is nonnegative and symmetric. Three-way generalizations of the triangle inequality and minimality from the literature are reviewed and it is investigated what forms of symmetry and nonnegativity are implied by the three-way axioms. A special form of three-way symmetry that can be deduced is equality of the diagonal planes of the three-dimensional cube. Furthermore, it is studied what diagonal plane equalities hold for the four-dimensional tesseract.


Diagonal plane equality Tetrahedron inequality Multi-way symmetry Three-way block Tesseract Multi-way dissimilarity 

Mathematics Subject Classification (2000)




The author thanks Hans-Hermann Bock and three anonymous reviewers for their helpful comments and valuable suggestions on earlier versions of this article.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution,and reproduction in any medium, provided the original author(s) and source are credited.


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Copyright information

© The Author(s) 2008

Authors and Affiliations

  1. 1.Psychometrics and Research Methodology Group, Leiden University Institute for Psychological ResearchLeiden UniversityLeidenThe Netherlands

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