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If $X$ is a smooth projective toric variety and $P \subset \mathbf{R}^n$ is its moment polytope, then a generic linear function on $\mathbf{R}^n$ induces (1) a Morse function on $X$, and (2) a shelling of the dual polytope $P^*$ ("line shelling").

The critical points of (1) map bijectively to the vertices of $P$, and are ordered by their critical values. They naturally match the top-dimensional faces of $P^*$ that are ordered by (2). The index of the $j$th critical point has an interpretation in terms of the shelling: it is the twice the cardinality of the $j$th "restriction set."

(It is easier to see what is Morse-index-like about this restriction set on $P$ instead of $P^*$. The linear function $f$ divides the set of edges incident with a vertex in $P$ into two: those with $f(e) \subset (-\infty,f(v)]$ and those with $f(e) \subset [f(v),\infty)$. The $j$th restriction set is naturally identified with the lower set of edges.)

A toric manifold, or any other compact manifold, has a self-indexing Morse function.

Can a simplicial polytope fail to have a "self-indexing shelling"? More precisely, a shelling where the restriction sets have cardinalities 0,1,1,1,..,1,2,2,..,2,3,3,..,3,4,... in that order?

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    $\begingroup$ This question is a strong version (because continuous deformations are not allowed) of the dual of mathoverflow.net/questions/199357. $\endgroup$ – Richard Stanley Mar 17 '15 at 19:14
  • $\begingroup$ Thanks Richard, I hadn't seen this question. Besides not allowing deformations, another difference is that I am curious about shellings that do not come from linear functions. $\endgroup$ – David Treumann Mar 17 '15 at 19:37

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