We introduce topological prismatoids, a combinatorial abstraction of the
(geometric) prismatoids recently introduced by the second author to construct
counter-examples to the Hirsch conjecture. We show that the `strong d-step
Theorem' that allows to construct such large-diameter polytopes from
`non-d-step' prismatoids still works at this combinatorial level. Then, using
metaheuristic methods on the flip graph, we construct four combinatorially
different non-d-step 4-dimensional topological prismatoids with 14
vertices. This implies the existence of 8-dimensional spheres with 18
vertices whose combinatorial diameter exceeds the Hirsch bound. These examples
are smaller that the previously known examples by Mani and Walkup in 1980 (24
vertices, dimension 11).
Our non-Hirsch spheres are shellable but we do not know whether they are
realizable as polytopes.Comment: 20 pages. Changes from v1 and v2: Reduced the part on shellability
and general improvement to accesibilit