The width of five‐dimensional prismatoids

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 d⩽4 . Motivated by this, we here study the width of five‐dimensional prismatoids, obtaining the following results: There are 5 ‐prismatoids of width 6 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 Ω(n) for arbitrarily large n . Hence, the width of 5 ‐prismatoids is unbounded. Both constructions, in particular that of a twenty‐dimensional non‐Hirsch polytope, are totally explicit.