Improved bounds for Hadwiger’s covering problem via thin-shell estimates
Bibliographic record
Abstract
A central problem in discrete geometry, known as Hadwiger's covering problem, asks what the smallest natural number N(n) is such that every convex body in \mathbb{R}^{n} can be covered by a union of the interiors of at most N(n) of its translates. Despite continuous efforts, the best general upper bound known for this number remains as it was more than sixty years ago, of the order of {2n \choose n}n\ln n . In this note, we improve this bound by a subexponential factor. That is, we prove a bound of the order of {2n \choose n}e^{-c\sqrt{n}} for some universal constant c>0 . Our approach combines ideas from [3] by Artstein-Avidan and the second named author with tools from asymptotic geometric analysis. One of the key steps is proving a new lower bound for the maximum volume of the intersection of a convex body K with a translate of -K ; in fact, we get the same lower bound for the volume of the intersection of K and -K when they both have barycenter at the origin. To do so, we make use of measure concentration, and in particular of thin-shell estimates for isotropic log-concave measures. Using the same ideas, we establish an exponentially better bound for N(n) when restricting our attention to convex bodies that are \psi_{2} . By a slightly different approach, an exponential improvement is established also for classes of convex bodies with positive modulus of convexity.
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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 | 0.006 | 0.030 |
| Meta-epidemiology (narrow) | 0.003 | 0.002 |
| Meta-epidemiology (broad) | 0.003 | 0.004 |
| Bibliometrics | 0.004 | 0.003 |
| Science and technology studies | 0.002 | 0.006 |
| Scholarly communication | 0.005 | 0.016 |
| Open science | 0.005 | 0.008 |
| Research integrity | 0.002 | 0.009 |
| Insufficient payload (model declined to judge) | 0.009 | 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".