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
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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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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 | 0.014 | 0.025 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.001 | 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 it