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 d ? 4. Motivated by this, we here study the width xf five-dimensional prismatoids, obtaining the following results: (i) 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. (ii) There are 5-prismatoids with n vertices and width ?(?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.

Autoría

Descargar referencia