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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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    $\begingroup$ Is the orientation coming from a generic linear functional? Without further assumptions, the fact that there is an overall minimum and maximum in your poset does not imply that each face has a minimum and a maximum. $\endgroup$ May 13, 2021 at 15:15
  • $\begingroup$ Ah, good point! I will edit my question: I meant that the condition "one source, one sink" is brutally required from all the faces as well. $\endgroup$ May 13, 2021 at 15:17
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    $\begingroup$ Note that the orientations from linear functionals will satisfy this condition (and are maybe a reasonable place to start thinking about it...) $\endgroup$ May 13, 2021 at 15:18
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    $\begingroup$ By the way, "thin" is a term of art in the area (see e.g. core.ac.uk/download/pdf/82377898.pdf), which might be confused for your term "slim." $\endgroup$ May 13, 2021 at 15:23
  • $\begingroup$ Oh, thanks for telling! Maybe I should call them "short" or "tight". $\endgroup$ May 13, 2021 at 15:29

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