I am from another area of maths, and has never learnt combinatorics and polytope theory properly before, so please feel free to point out any mistake/misuse of terminology.
The starting point is the following ``dimension 1 (or 2)" situation: Given a convex (regular) $n$-gon with vertices labelled $1,2,\ldots,n$. A noncrossing partition is a partition of these vertices, so that for any pairs of subclasses, the corresponding convex hulls do not intersect (i.e. non-crossing). For example, if $n=3$, we have $\{1,2,3\}, \{\{1\},\{2\},\{3\}\}, \{\{1,2\},\{3\}, \{\{1,3\},\{2\}\}, \{\{2,3\},\{1\}\}$. It is well-known that the number of non-crossing partitions for a convex $n$-gon is counted by the Catalan number.
I want to see if there is any ``higher dimensional" analogue for non-crossing partition, where $n$-gon (2-dimensional cyclic polytope with $n$ vertices) in is replaced by $2d$-dimensional cyclic polytope with $n$ vertices. Does anyone know if such theory exists?