Faces of Polytopes

Abstract

The first two chapters of this thesis are about face numbers of polytopes. In Chapter 1, we settle a question of B\'ar\'any from the 1990s: do all convex $d$-polytopes satisfy $f_0, \ldots, f_{d-1} \geq \min\{f_0,f_{d-1}\}$? We answer ``yes" and prove a stronger statement: for all convex $d$-polytopes and $k=0,\ldots,d-1$,\[ f_k \geq \frac{1}{2}\left[ \binom{\lceil\frac{d}{2}\rceil}{k} + \binom{\lfloor\frac{d}{2}\rfloor}{k} \right]f_0,\qquad f_k \geq \frac{1}{2}\left[ \binom{\lceil\frac{d}{2}\rceil}{d-k-1} + \binom{\lfloor\frac{d}{2}\rfloor}{d-k-1} \right]f_{d-1}. \] The former holds with equality precisely when $k=0$ or for simple polytopes when $k=1$. The latter holds with equality precisely when $k=d-1$ or for simplicial polytopes when $k=d-2$. These inequalities are asymptotically tight for dual cyclic and cyclic polytopes, respectively. We generalize the latter inequality to shellable, strongly regular CW spheres and balls, proving that every shellable, dual shellable, strongly regular CW $(d-1)$-sphere satisfies $f_0, \ldots, f_{d-1} \geq \min\{f_0,f_{d-1}\}$. Chapter 2 concerns the \emph{fatness} parameter, defined for 4-polytopes and 3-spheres as\[ \frac{f_1+f_2-20}{f_0+f_3-10}. \] We construct a family of shellable, dual shellable, strongly regular CW 3-spheres with arbitrarily high fatness, solving a problem of Ziegler. These spheres have $f$-vectors $(\Theta(n),\allowbreak\Theta(n\alpha(n)),\allowbreak\Theta(n\alpha(n)),\allowbreak\Theta(n))$, where $\alpha$ is the inverse Ackermann function. We conjecture that these spheres are realizable as convex 4-polytopes; if true, this would answer Eppstein, Kuperberg, and Ziegler's famous question of whether 4-polytopes may be arbitrarily fat. Chapter 3 is about reconstruction problems. We prove that every 4-polytope can be reconstructed up to combinatorial isomorphism from its edge-polygon incidences, solving a problem of Gr\"unbaum. For each $d \geq 3$, we prove that \emph{not} every $d$-polytope can be reconstructed up to combinatorial isomorphism from its $(d-3)$-skeleton and dual $(d-3)$-skeleton together, answering a question of Samper. As a corollary, we fully characterize the subsets $K\subseteq\{0,\ldots,d-1\}$ such that every $d$-polytope can be reconstructed from the incidences of its faces of dimension in $K$. We further prove positive and negative reconstruction results on simplicial homology manifolds and pseudomanifolds.