Bibliographic record
Abstract
Abstract This paper studies a new Whitney type inequality on a compact domain $\Omega \subset {\mathbb R}^d$ that takes the form $$ \begin{align*} \inf_{Q\in \Pi_{r-1}^d(\mathcal{E})} \|f-Q\|_p \leq C(p,r,\Omega) \omega_{\mathcal{E}}^r(f,\mathrm{diam}(\Omega))_p,\ \ r\in {\mathbb N},\ \ 0 where $\omega _{\mathcal {E}}^r(f, t)_p$ denotes the rth order directional modulus of smoothness of $f\in L^p(\Omega )$ along a finite set of directions $\mathcal {E}\subset \mathbb {S}^{d-1}$ such that $\mathrm {span}(\mathcal {E})={\mathbb R}^d$ , $\Pi _{r-1}^d(\mathcal {E}):=\{g\in C(\Omega ):\ \omega ^r_{\mathcal {E}} (g, \mathrm {diam} (\Omega ))_p=0\}$ . We prove that there does not exist a universal finite set of directions $\mathcal {E}$ for which this inequality holds on every convex body $\Omega \subset {\mathbb R}^d$ , but for every connected $C^2$ -domain $\Omega \subset {\mathbb R}^d$ , one can choose $\mathcal {E}$ to be an arbitrary set of d independent directions. We also study the smallest number $\mathcal {N}_d(\Omega )\in {\mathbb N}$ for which there exists a set of $\mathcal {N}_d(\Omega )$ directions $\mathcal {E}$ such that $\mathrm {span}(\mathcal {E})={\mathbb R}^d$ and the directional Whitney inequality holds on $\Omega $ for all $r\in {\mathbb N}$ and $p>0$ . It is proved that $\mathcal {N}_d(\Omega )=d$ for every connected $C^2$ -domain $\Omega \subset {\mathbb R}^d$ , for $d=2$ and every planar convex body $\Omega \subset {\mathbb R}^2$ , and for $d\ge 3$ and every almost smooth convex body $\Omega \subset {\mathbb R}^d$ . For $d\ge 3$ and a more general convex body $\Omega \subset {\mathbb R}^d$ , we connect $\mathcal {N}_d(\Omega )$ with a problem in convex geometry on the X-ray number of $\Omega $ , proving that if $\Omega $ is X-rayed by a finite set of directions $\mathcal {E}\subset \mathbb {S}^{d-1}$ , then $\mathcal {E}$
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.003 | 0.006 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.001 | 0.005 |
| Insufficient payload (model declined to judge) | 0.015 | 0.001 |
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 source (direct Gemma or distilled Codex), 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".