The Riesz $α$-energy of log-concave functions and related Minkowski problem
Bibliographic record
Abstract
We calculate the first order variation of the Riesz $α$-energy of a log-concave function $f$ with respect to the Asplund sum. Such a variational formula induces the Riesz $α$-energy measure of log-concave function $f$, which will be denoted by $\mathfrak{R}_α(f, \cdot)$. We pose the related Riesz $α$-energy Minkowski problem aiming to find necessary and/or sufficient conditions on a pregiven Borel measure $μ$ defined on $\Rn$ so that $μ=\mathfrak{R}_α(f,\cdot)$ for some log-concave function $f$. Assuming enough smoothness, the Riesz $α$-energy Minkowski problem reduces to a new Monge-Ampère type equation involving the Riesz $α$-potential. Moreover, this new Minkowski problem can be viewed as a functional counterpart of the recent Minkowski problem for the chord measures in integral geometry posed by Lutwak, Xi, Yang and Zhang (Comm.\ Pure\ Appl.\ Math.,\ 2024). The Riesz $α$-energy Minkowski problem will be solved under certain mild conditions on $μ$.
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 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".