# What is known about the combinatorics of the hyperplane arrangement spanned by cyclic polytopes?

Let $$1\leq d$$ be an integer.

Consider the $$d$$-dimensional moment curve $$\mu\colon \mathbb R\to \mathbb R^d$$ given by $$t\mapsto (t,t^2,\dots, t^d)$$. Given a finite subset $$S\subset \mathbb R$$ of cardinality $$\geq d+1$$, the $$d$$-dimensional cyclic polytope $$C(d,S)$$ is the convex hull of $$\mu(S)$$ in $$\mathbb R^d$$.

It is well known, that the combinatorial type of the polytope $$C(d,S)$$ does depend (for fixed $$d$$) only on the cardinality of $$S$$. Cyclic polytopes and their triangulations can be studied by varying $$d$$ and exploiting the relationships arising from the obvious projection map $$C(d+1,S)\to C(d,S)$$ (forgetting the last coordinate); see e.g. Rambau's thesis.

The cyclic polytope $$C(d,S)$$ can be equipped with additional structure as follows: every subset $$T\subset S$$ of cardinality $$d$$ determines an affine hyperplane spanned by $$\mu(T)$$ in $$\mathbb R^d$$; all together this gives a hyperplane arrangement $$H(d,S)$$. If we restrict this hyperplane arrangement to the cyclic polytope, we get a partition $$P(d,S)$$ of $$C(d,S)$$ into convex pieces with pairwise disjoint interiors.

What is known about the combinatorics of the hyperplane arrangement $$H(d,S)$$ and about the corresponding partition $$P(d,S)$$? Is there a systematic study a la Rambau?

Note that $$H(d,S)$$ is no longer invariant if one replaces $$S$$ by a different set of the same cardinality. Let us denote by $$H(d,n)$$ the "standard" arrangement where $$S=[n]=\{0,\dots, n\}$$.

For which $$S$$ is $$H(d,S)$$ combinatorially equivalent to $$H(d,n)$$? For which $$S$$ is $$P(d,S)$$ combinatorially equivalent to $$P(d,n)$$?

Finally, let me ask a more specific question which is motivated by some pictures I drew in dimension two. Given a surjective weakly monotone map $$f\colon[n]\to[d]$$, let $$U_f$$ denote the collection of those $$d$$-dimensional simplices $$\Delta^I=C(d,I)$$ spanned by the vertices $$\mu(I)$$ for some $$I\subset [n]$$ such that $$f|I\colon I\to [d]$$ is a bijection. My two-dimensional pictures suggest:

Is it true that, for every $$f$$ as above, the polytope $$\bigcap U_f \subset C(d,n)$$ is (A) a piece of the partition $$P(d,n)$$ and (B) a $$d$$-dimensional simplex?

• See arxiv.org/abs/1608.08288, but I think this notion of "cyclic hyperplane arrangement" is different than what you're talking about. – Sam Hopkins Feb 13 at 15:08