2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/65910We make explict a description in terms of convex geometry of the higher Bruhat orders B(n,d) sketched by Kapranov and Voevodsky. We give an analogous description of the higher Stasheff-Tamari poset S_1(n,d). We show that the map f sketched by Kapranov and Voevodsky from B(n,d) to S([0,n+1],d+1) coincides with the map constructed by Rambau, and is a surjection for d<=2. We construct a map analogous to f from S_1(n,d) to B(n-1,d), and show that it is always a poset embedding. We also give an explicit criterion to determine if an element of B(n-1,d) is in the image of this map.26 pages, 5 figures. Changes from version one are minor and confined to one paragraphCombinatoricsMetric Geometry52B05 (Primary); 06A07; 52B12; 57Q15 (Secondary)text