On the Diameter of Polytopes
On the Diameter of Polytopes

Benjamin Matschke, IAS
Fine Hall 224
Santos' construction of counterexamples to the Hirsch conjecture is based on the existence of prismatoids of dimension $d$ of width greater than $d$. The case $d=5$ being the smallest one in which this can possibly occur, we here study the width of 5dimensional prismatoids, obtaining the following results:  There are 5prismatoids of width six with only 25 vertices, versus the 48 vertices in Santos' original construction. This leads to lowering the dimension of the nonHirsch polytopes from 43 to only 20.  There are 5prismatoids with n vertices and $width \Omega(n^(1/2))$ for arbitrarily large $n$.
This is joint work with Francisco Santos and Christophe Weibel.