Anders Björner: Interlacing of toric h-vectors of Schlegel diagrams

A Schlegel $(d-1)$-diagram is a polytopal subdivision of a $(d-1)$-polytope, obtained by projecting the boundary complex of a $d$-polytope $P$ onto one of its facets. It is used to investigate $P$ by visualization in a lower dimension.

The toric $h$-vector $h=(h_0, h_1, \ldots, h_d)$ is recursively defined for any $(d-1)$-dimensional polytopal complex, in particular for Schlegel diagrams and other subdivisions of balls. In the simplicial case it is the $h$-vector well known from the theory of Stanley-Reisner rings.

We prove that the toric $h$-vector of a Schlegel $(d-1)$-diagram satisfies $$ h_{d} \le h_{0} \le h_{d-1} \le h_{1} \le h_{d-2} \le h_{2} \le \cdots \le h_{\lfloor d/2\rfloor+1} \le h_{\lfloor d/2\rfloor-1} \le h_{\lfloor d/2\rfloor} $$