On Homogeneous Convex Cones, The Carathéodory Number, and the Duality Mapping
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Bibliographic record
Abstract
Using three simple examples, we answer three questions related to homogeneous convex cones, the Carathéodory number of convex cones, and self-concordant barriers for convex cones. First, we show that, if the convex cone is not homogeneous, then the duality mapping does not have to be an involution. Next, we show that there are very elementary convex cones that are not homogeneous but have invariant Carathéodory number in the interior. Third, we show that the invariance of the Carathéodory number in the interior of the convex cone does not suffice to make the cone homogeneous even if the cone is self-dual. Finally, we characterize the Carathéodory number of epigraphs of matrix norms.
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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 | 0.006 | 0.006 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| 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