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.


62A01 Foundations and philosophical topics in statistics
51M16 Inequalities and extremum problems in real or complex geometry
62H99 Multivariate analysis
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