  We 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.
