Filling metric spaces
Bibliographic record
Abstract
We prove a new version of isoperimetric inequality: Given a positive real m, a Banach space B, a closed subset Y of metric space X, and a continuous map f:Y→B with f(Y) compact infFHCm+1(F(X))≤c(m)HCm(f(Y))m+1 m, where HCm denotes the m-dimensional Hausdorff content, the infimum is taken over the set of all continuous maps F:X⟶B such that F(y)=f(y) for all y∈Y, and c(m) depends only on m. Moreover, one can find F with a nearly minimal HCm+1 such that its image lies in the C(m)HCm(f(Y))1 m-neighborhood of f(Y) with the exception of a subset with zero (m+1)-dimensional Hausdorff measure. The paper also contains a very general coarea inequality for Hausdorff content and its modifications. As an application we demonstrate an inequality conjectured by Larry Guth that relates the m-dimensional Hausdorff content of a compact metric space with its (m−1)-dimensional Urysohn width. We show that this result implies new systolic inequalities that both strengthen the classical Gromov’s systolic inequality for essential Riemannian manifolds and extend this inequality to a wider class of non-simply-connected manifolds.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 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".