On multi-way metricity, minimality and diagonal planes

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.