More bounds on the diameters of convex polytopes
Bibliographic record
Abstract
Finding a good bound on the maximal edge diameter $Δ(d,n)$ of a polytope in terms of its dimension $d$ and the number of its facets $n$ is one of the basic open questions in polytope theory \cite{BG}. Although some bounds are known, the behaviour of the function $Δ(d,n)$ is largely unknown. The Hirsch conjecture, formulated in 1957 and reported in \cite{GD}, states that $Δ(d,n)$ is linear in $n$ and $d$: $Δ(d,n) \leq n-d$. The conjecture is known to hold in small dimensions, i.e., for $d \leq 3$ \cite{VK}, along with other specific pairs of $d$ and $n$ (Table \ref{before}). However, the asymptotic behaviour of $Δ(d,n)$ is not well understood: the best upper bound -- due to Kalai and Kleitman -- is quasi-polynomial \cite{GKDK}. In this article we will show that $Δ(4,12)=7$ and present strong evidence for $Δ(5,12)=Δ(6,13)=7$. The first of these new values is of particular interest since it indicates that the Hirsch bound is not sharp in dimension 4.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.017 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.003 |
| Bibliometrics | 0.004 | 0.003 |
| Science and technology studies | 0.002 | 0.005 |
| Scholarly communication | 0.005 | 0.014 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.002 | 0.009 |
| Insufficient payload (model declined to judge) | 0.015 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".