The positive Dressian equals the positive tropical Grassmannian

The Dressian and the tropical Grassmannian parameterize abstract realizable linear spaces; but in general, is much larger than Grassmannian. There are natural positive notions of both these spaces -- Dressian, (which we introduced roughly fifteen years ago) so it to ask how two compare. In this paper show that equals Using connection between regular positroidal subdivisions hypersimplex, use our result give a new proof da Silva's 1987 conjecture (first proved 2017 by Ardila-Rincon-Williams) all positively oriented matroids realizable. We also finest hypersimplex consist series-parallel matroid polytopes, achieve equality Speyer's f-vector theorem. Finally an example subdivision which not regular, make theory hyperplane arrangements.