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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?

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

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