Here is a link for the just-posted paper Diameter of Polyhedra: The Limits of Abstraction by Freidrich Eisenbrand, Nicolai Hahnle, Sasha Razborov, and Thomas Rothvoss.
And here is a link to the paper by Sandeep Koranne and Anand Kulkarni “The d-step Conjecture is Almost true” – most of the discussion so far was in this direction.
We had a long and interesting discussion regarding the Hirsch conjecture and I would like to continue the discussion here.
The way I regard the open collaborative efforts is as an open collective attempt to discuss and make progress on the problem (and to raise more problems), and also as a way to assist people who think or work (or will think or will work) on these problems on their own.
Most of the discussion in the previous thread was not about the various problems suggested there but rather was about trying to prove the Hirsch Conjecture precisely! In particular, the approach of Sandeep Koranne and Anand Kulkarni which attempts to prove the conjecture using “flips” (closely related to Pachner moves, or bistaller operations) was extensively discussed. Here is the link to another paper by Koranne and Kulkarni “Combinatorial Polytope Enumeration“. There is certainly more to be understood regarding flips, Pachner moves, the diameter, and related notions. For example, I was curious about for which Pachner moves “vertex decomposibility” (a strong form of shellability known to imply the Hirsch bound) is preserved. We also briefly discussed metric aspects of the Hirsch conjecture and random polytopes.
For general background: Here is a chapter that I wrote about graphs, skeleta and paths of polytopes. Some papers on polytopes on Gunter Ziegler’s homepage describe very interesting and possibly relevant current research in this area. Here is a link to Eddie Kim and Francisco Santos’s survey article on the Hirsch Conjecture.
Here is a link from the open problem garden to the continuous analog of the Hirsch conjecture proposed by Antoine Deza, Tamas Terlaky, and Yuriy Zinchenko.