This is a "looking for context" question.

Let's say that a polytope is directed if its 1-skeleton is an oriented graph with no cycles, one source, one sink. (Edit: let us additionally assume that every face also has one source, one sink.) In this case, the vertices form a poset, and we can extend this poset structure to a (non-reflexive) relation on all the faces, by saying $F_1 < F_2$ if $\max (F_1) \leq \min (F_2)$. Then we can consider face-chains with respect to this new relation (notice that a vertex is comparable with itself so a vertex can repeat in a chain, while bigger faces can't). For a face-chain $(F_1 < \ldots < F_n)$ inside the face $F$, let's say that its *dimension gain* is $(\sum \dim F_i - \dim F)$. Note that dimension gains can be both positive and negative. Now, the definition:

**Definition.** A directed polytope is *slim* if a chain of length $n>1$ has dimension gain at most $n-2$.

**Examples.** Simplices and cubes with standard directions are slim. Associahedra with Tamari directions are slim.

Has this condition appeared in the literature previously?

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