UW Combinatorics and Geometry Seminar: Josh Hinman

We generalize a result about the face numbers of polytopes to the realm of CW spheres. Let X be a CW sphere such that: 

• X is strongly regular. (The intersection of any two faces is a face.) 
• X is shellable. (We can build X, facet by facet, so that each facet intersects with the previous ones in a predictable way.)

Then X is a "squishy polytope", resembling a polytope topologically and combinatorially but not geometrically. We prove a set of linear lower bounds on the face numbers of X in terms of the number of facets. To do so, we translate an argument about solid angles into an argument about shellings.