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 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) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 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".