On the Number of Tetrahedra with Minimum, Unit, and Distinct Volumes in Three-Space
Bibliographic record
Abstract
We formulate and give partial answers to several combinatorial problems on volumes of simplices determined by n points in 3-space, and in general in d dimensions. (i) The number of tetrahedra of minimum (non-zero) volume spanned by n points in $\mathbb{R}$ 3 is at most $\frac{2}{3}n^3-O(n^2)$ , and there are point sets for which this number is $\frac{3}{16}n^3-O(n^2)$ . We also present an O ( n 3 ) time algorithm for reporting all tetrahedra of minimum non-zero volume, and thereby extend an algorithm of Edelsbrunner, O'Rourke and Seidel. In general, for every $k,d\in \mathbb{N}, 1\leq k \leq d$ , the maximum number of k -dimensional simplices of minimum (non-zero) volume spanned by n points in $\mathbb{R}$ d is Θ( n k ). (ii) The number of unit volume tetrahedra determined by n points in $\mathbb{R}$ 3 is O ( n 7/2 ), and there are point sets for which this number is Ω( n 3 log log n ). (iii) For every $d\in \mathbb{N}$ , the minimum number of distinct volumes of all full-dimensional simplices determined by n points in $\mathbb{R}$ d , not all on a hyperplane, is Θ( n ).
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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 | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 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".