A study of perimeters for a class of triangles contained in the unit ball of normed spaces
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Bibliographic record
Abstract
Recently, A. Ahmad, Y. Fu and Y. Li considered a class of triangles inscribed in the unit ball of a Banach space; they introduced two parameters, based on the perimeter of these triangles, and indicated a few results. Here we deepen their study, we give several new results and we compare these parameters with other ones. We consider triangles $T(x,y,z)$ with $x,y,z$ in the unit sphere and such that $x+y+z=0$. We compare this class with the one of equilateral triangles, studied in a recent paper by J. Alonso, P. Mart{\'i}n and P.L. Papini. We shall use also the modulus of convexity and the modulus of smoothness to give some estimates concerning our parmeters. We also indicate some open problems.
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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) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| 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 it