On the <i>L</i> <i>p</i> Brunn-Minkowski theory and the <i>L</i> <i>p</i> Minkowski problem for <i>C</i>-coconvex sets
Bibliographic record
Abstract
Abstract Let $C$ be a pointed closed convex cone in ${{\mathbb {R}}^n}$ with vertex at the origin $o$ and having nonempty interior. The set $A\subset C$ is $C$-coconvex if the volume of $A$ is finite and $A^{\bullet }=C\setminus A$ is a closed convex set. For $0&lt;p&lt;1$, the $p$-co-sum of $C$-coconvex sets is introduced and the corresponding $L_p$ Brunn–Minkowski inequality for $C$-coconvex sets is established. We also define the $L_p$ surface area measures, for $0\neq p\in {\mathbb {R}}$, of certain $C$-coconvex sets, which are critical in deriving a variational formula of the volume of the Wulff shape associated with a family of functions obtained from the $p$-co-sum. This motivates the $L_p$ Minkowski problem aiming to characterize the $L_p$ surface area measures of $C$-coconvex sets. The existence of solutions to the $L_p$ Minkowski problem for all $0\neq p\in {\mathbb {R}}$ is established. The $L_p$ Minkowski inequality for $0&lt;p&lt;1$ is proved and is used to obtain the uniqueness of the solutions to the $L_p$ Minkowski problem for $0&lt;p&lt;1$. For $p=0$, we introduce $(1-\tau )\diamond A_1\oplus _0\tau \diamond A_2$, the log-co-sum of two $C$-coconvex sets $A_{1}$ and $A_{2}$ with respect to $\tau \in (0, 1)$, and prove the log-Brunn–Minkowski inequality of $C$-coconvex sets. The log-Minkowski inequality is also obtained and is applied to prove the uniqueness of the solutions to the log-Minkowski problem that characterizes the cone-volume measures of $C$-coconvex sets. Our result solves an open problem raised by Schneider [41].
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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 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.005 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 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 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".