# Mathematics Colloquia and Seminars

The following three classes contain very similar objects --- in a topological resp. geometric resp. combinatorial model: \begin{itemize}\itemsep=0pt \item $3$-dimensional CW-spheres with the intersection property, \item $4$-dimensional convex polytopes, and \item Eulerian lattices of rank~$5$. \end{itemize} We introduce and study the parameter of \emph{fatness}, $\frac{f_1+f_2}{f_0+f_3}$ for these three classes --- which seems to be a key indicator to show how little we know. So, it is not clear whether fatness is bounded at all on any of these classes. Here we construct examples of \begin{itemize}\itemsep=0pt \item rational $4$-dimensional convex polytopes of fatness larger than $5-\varepsilon$, \item $4$-dimensional convex polytopes of fatness larger than $5.01$, and \item $3$-dimensional CW-spheres with the intersection property of fatness larger than $6-\varepsilon$. \end{itemize} This implies counter-examples to conjectured $f$-vector inequalities of Bayer (1988) and of Billera \& Ehrenborg (1999). Most of our examples are constructed using the Eppstein construction'': as the convex hull of a $4$-polytope with all ridges tangent to~$S^3$, and its polar. This construction has a close connection with ball packings in~$S^3$. Their study should lead to an infinite family of $2$-simple $2$-simplicial polytopes. (Joint work with David Eppstein, UC Irvine)