Points with large quadrant depth
Bibliographic record
Abstract
Given a set P of points in the plane we are interested in points that are `deep' in the set in the sense that they have two opposite quadrants both containing many points of P. We deal with an extremal version of this problem. A pair (a,b) of numbers is admissible if every point set P contains a point p in P that determines a pair (Q,Qop) of opposite quadrants, such thatQ contains at least an a-fraction and Qop contains at least a b-fraction of the points of P. We provide a complete description of the set F of all admissible pairs (a,b). This amounts to identifying three line segments and a point on the boundary of F. In higher dimensions we study the maximum a, such that (a,a) is opposite-orthant admissible. In dimension d we show that 1/(2γ)≤a≤1/γ for γ=22d-12d-1. Finally we deal with a variant of the problem where the opposite pairs of orthants need not be determined by a point in P. Again we are interested in values a, such that all subsets P inRd admit a pair (O,Oop) of opposite orthants both ontaining at least an a-fraction of the points. The maximum such value is a=1/2d. Generalizations of the problem are also disussed.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 teacher head, 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".