Th Lp John ellipsoids for general measures
Bibliographic record
Abstract
This thesis aims to develop the Lp John ellipsoids related to general measures. Our Lp John ellipsoids contain many well-known ellipsoids constructed from given convex bodies as special cases, including but not limited to the classical John ellipsoid, the Lp John ellipsoid, the Lutwak-Yang-Zhang ellipsoid, the Petty ellipsoid, etc. Let μ be an α-homogeneous measure on Rn for α > 0. Our Lp John ellipsoids for the general measure μ for p > 0 are defined as the solutions to the following optimization problem: max V(E) subject to Vμ,p(K,E)≥μ(K), E∈ε₀n where E₀n denotes the set of all origin-symmetric ellipsoids, K is a compact convex set in Rn containing the origin in its interior, V is the volume function, and Vμ,p(K,E) = 1/αμ(K) [integral of] Sn⁻¹ hᴾE(v)dSμ,p(K,v), with hE the support function of E and dSμ,p(K,v) the Lp-surface μ-area measure of K. In this thesis, for p > 0, we establish the existence and uniqueness of the Lp John ellipsoid for μ. A characterization of the Lp John ellipsoid for μ is obtained. We also investigate the case for p = 0, which is related to the logarithmic function. Besides, the inclusion for the Lp John ellipsoid for μ is provided. The convex bodies with identical John and Lp John ellipsoids for the general measure μ are characterized. Finally, we provide a study for another arguably more general family of Lp John ellipsoids, defined in a way similar to the one in (1) but with Vμ,p(K,E) replaced by [integral of] Sn⁻¹ hᴾE(v)dv(v) and with μ(K) replaced by v(Sn⁻¹), respectively.
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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 | 0.003 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.005 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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".