THE STEINER CENTRE OF A SET OF POINTS: STABILITY, ECCENTRICITY, AND APPLICATIONS TO MOBILE FACILITY LOCATION
Bibliographic record
Abstract
The Euclidean centre (centre of the smallest enclosing sphere) of a set of points P in two or more dimensions is unstable; small perturbations at only a few points of P can result in an arbitrarily large relative change in the position of the Euclidean centre. Any centre function more stable than the Euclidean centre is eccentric; that is, its associated radius exceeds the radius of the smallest enclosing circle for some point sets P. Motivated by applications in mobile facility location (in which clients move continuously with some maximum velocity) we seek alternative notions of centrality that are stable while maintaining low eccentricity. In general there is a trade-off; centre functions with lower eccentricity are less stable. In an attempt to balance the conflicting goals of closeness of approximation and stability, we apply the Steiner centre, traditionally defined for convex polytopes, as a centre function of a set of points in the plane. Although previously defined, the notion of a Steiner centre had not been analyzed in terms of its approximation of the Euclidean centre. Exploiting the equivalence of the two definitions of the Steiner centre established by Shephard,27we prove the stability of the Steiner centre is π/4 and show that the associated radius is at most 1.1153 times the Euclidean radius of any point set P. It follows that a mobile facility located at the Steiner centre of the positions of a set of mobile clients remains close to the Euclidean centre of the clients yet never moves with relative velocity that exceeds 4/π.
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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 | 0.019 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.003 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 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".