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The secondary fan (k, n) is a polyhedral fan which stratifies the regular subdivisions of the hypersimplices (k, n). We find new infinite families of rays of (k, n), and we compute the fans (2, 7) and (3, 6). In the special case k=2 the fan (2, n) is closely related to the metric fan MF (n), which forms a natural parameter space for the metric spaces on n points. So our results yield a classification of the finite metric spaces on seven points.
Casabella et al. (2024) studied this question.