The Existence of Convex Body with Prescribed Curvature Measures
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Bibliographic record
Abstract
Curvature measure and surface area measure are the basic notions in the classical differential geometry. They play fundamental roles in the theory of convex bodies. They are closely related to the differential geometry and integral geometry of convex hypersurfaces. The Minkowski problem is the problem of prescribing nth surface area measure on . The Christoffel problem concerns the prescribing the first surface area measure (e.g. see [1, 3, 6, 7, 14, 17, 19]). The general problem of prescribing surface area measures is called the Christoffel–Minkowski problem, we refer [12] for an updated account. The problem of prescribing zeroth curvature measure is called the Alexandrov problem, which is a counterpart to Minkowski problem. The problem is equivalent to solve a Monge–Ampère-type equation on . The existence and uniqueness were obtained by Alexandrov [2]. The regularity of the Alexandrov problem in elliptic case was proved by Pogorelov [18] for n = 2 and by Oliker [16] for higher-dimension case. The general regularity results (degenerate case) of the problem were obtained in [9]. The general problem of prescribing (n − k)th curvature measure for case k ⩽ n is an interesting counterpart of the Christoffel–Minkowski problem. It has been discussed in literature (e.g. [20]). Nevertheless, very little is known except for the Alexandrov problem.
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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 | 0.003 | 0.007 |
| 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 | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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