A High-Dimensional Diameter Problem for Polytopes

Avi Wigderson is here for a year and it was a good opportunity to go back together to the question of diameter of polytopes. The diameter problem for polytopes is to determine the behavior of the maximum diameter of the graph of a simple -polytopes with facets. We gave this problem considerable attention over here. (For example, we devoted to it polymath3.)

Let me mention an interesting problem about a high dimensional generalization of the “polynomial Hirsh conjecture”.

Let be a simple -polytope with facets (or, more generally, the dual of a simplicial -sphere with vertices).

Is the following true:

Every polygon in the graph of with edges is the boundary of a 2-chain with norm (sum of coefficients) at most a polynomial number in , and ?

One can be more ambitious and ask for a a representation of as the boundary of a polyhedral 2-ball . (In this case the norm is simply the number of -faces of .)

But I am not sure even about this question:

Does every induced polygon in a graph of a simple polytope the boundary of some 2-ball (in the 2-skeleton of )?

Is it known?

Remarks: a) Of course we can consider the question for cycles of dimensions greater than . We want to understand the behavior of the function such that for every simple -polytope with facets we can represent every dimensional cycle with , as the boundary of a -dimensional chain ,  such that

.

b) One can ask also for an analog for the revisiting path question. And c) restriction attention to (duals) of vertex-decomposable spheres also is reasonable. This question is related to systolic inequalities and to high-dimensional expanders. As always, comments are welcome (also regarding polymath3.)