Abstract

Santos’ construction of counter-examples to the Hirsch Conjecture (2012) is based on the existence of prismatoids of dimension d of width greater than d. Santos, Stephen and Thomas (2012) have shown that this cannot occur in dimension less than 5. Motivated by this we here study the width of 5-dimensional prismatoids, obtaining the following results:

- There are 5-prismatoids of width six with only 25 vertices, versus the 48 vertices in Santos’ original construction. This leads to non-Hirsch polytopes of dimension 20, rather than the original dimension 43.

- There are 5-prismatoids with n vertices and width Omega( qrt{n}) for arbitrarily large n. Hence, the width of 5-prismatoids is unbounded.

This is joint work with Francisco Santos and Christophe Weibel.