Minimizing the mean projections of finite $\rho$-separable packings
Bibliographic record
Abstract
A packing of translates of a convex body in the $d$-dimensional Euclidean\nspace $\\mathbb{E}^d$ is said to be totally separable if any two packing\nelements can be separated by a hyperplane of $\\mathbb{E}^{d}$ disjoint from the\ninterior of every packing element. We call the packing $\\mathcal P$ of\ntranslates of a centrally symmetric convex body $\\mathbf{C}$ in $\\mathbb{E}^d$\na $\\rho$-separable packing for given $\\rho\\geq 1$ if in every ball concentric\nto a packing element of $\\mathcal P$ having radius $\\rho$ (measured in the norm\ngenerated by $\\mathbf{C}$) the corresponding sub-packing of $\\mathcal P$ is\ntotally separable. The main result of this paper is the following theorem.\nConsider the convex hull $\\mathbf{Q}$ of $n$ non-overlapping translates of an\narbitrary centrally symmetric convex body $\\mathbf{C}$ forming a\n$\\rho$-separable packing in $\\mathbb{E}^d$ with $n$ being sufficiently large\nfor given $\\rho\\geq 1$. If $\\mathbf{Q}$ has minimal mean $i$-dimensional\nprojection for given $i$ with $1\\leq i<d$, then $\\mathbf{Q}$ is approximately a\n$d$-dimensional ball. This extends a theorem of K. B\\"or\\"oczky Jr. [Monatsh.\nMath. 118 (1994), 41-54] from translative packings to $\\rho$-separable\ntranslative packings for $\\rho\\geq 1$.\n
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 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 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".