Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Anyway, I edited the description. Hope this makes things clear for those who don't see any mathematics in the question. I'm certainly not expecting answers like "we are ignorant" or "maths is beautiful".
@kantelope : I don't see how this is not a mathematics question. It is about a new mathematical discovery, which contributes to the solution of a long standing mathematical problem. The question ask about the things we can learn from it, which could be new insights, new techniques, etc. For those who are interested in the problem, this should be the first thing: look at the results and get inspired. This makes sense even before the detail is published.
I would rephrase the question as follows: can we always find a convex polytope $P'$ such that $P'$ is combinatorially equivalent to $P$ and $P'$ is index increasing w.r.t. a given vector (after all, we can fix the vector wlog)
@TimothyChow, Thank you for this information. I remember that one. But here $\epsilon$ is supposed to be arbitrarily small. I think my second formulation is better.
