# "Slim" directed polytopes: any established name for them?

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?

• 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. May 13, 2021 at 15:15
• 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. May 13, 2021 at 15:17
• Note that the orientations from linear functionals will satisfy this condition (and are maybe a reasonable place to start thinking about it...) May 13, 2021 at 15:18
• 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." May 13, 2021 at 15:23
• Oh, thanks for telling! Maybe I should call them "short" or "tight". May 13, 2021 at 15:29