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We introduce segmented pseudometrics on a finite dimensional vector space as those having triangle equality at any intermediate point of any segment. Some example families are given. We characterize when a segmented (pseudo)metric is (semi)norm induced. We investigate the relation between a pseudometric being segmented and the concave four-point Fermat-Torricelli property, which states that the sum of the four pseudometric distances to the vertices of a triangle and to some inner point is always minimized at this inner point. These two properties are shown to be equivalent in dimension at most 2. For metrics in higher dimension the second property always implies the first, but not inversely, even when induced by a norm.

Segmented pseudometrics and four-point Fermat-Torricelli problems