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**On multi-way metricity, minimality and diagonal planes.**
*(English)*
Zbl 1306.62037

Summary: 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.

### MSC:

62A01 | Foundations and philosophical topics in statistics |

51M16 | Inequalities and extremum problems in real or complex geometry |

62H99 | Multivariate analysis |

### Keywords:

diagonal plane equality; tetrahedron inequality; multi-way symmetry; three-way block; tesseract; multi-way dissimilarity
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\textit{M. J. Warrens}, Adv. Data Anal. Classif., ADAC 2, No. 2, 109--119 (2008; Zbl 1306.62037)

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### References:

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