n-Semimetrics

We introduce n -semimetrics as a common extension of n -metrics and certain recent applied notions like the 3-way distance. The n -semimetrics are totally symmetric maps fromEn + 1 into R + satisfying the simplex inequality, a direct extension of the common triangle inequality. Among the examples, we study in detail certain n -semimetrics on {0, 1}m. We give a few constructions and extend the 2-way distances.