On a Duality between Metrics and $Σ$-Proximities
| dc.creator | Chebotarev, P. Yu. | |
| dc.creator | Shamis, E. V. | |
| dc.date | 2005-08-10 | |
| dc.date.accessioned | 2026-07-07T06:42:47Z | |
| dc.date.available | 2026-07-07T06:42:47Z | |
| dc.description | : In studies of discrete structures, functions are frequently used that express proximity, but are not metrics. We consider a class of such functions that is characterized by a normalization condition and an inequality that plays the same role as the triangle inequality does for metrics. We show that the introduced functions, named $Σ$-proximities, are in a definite sense dual to metrics: there exists a natural one-to-one correspondence between metrics and $Σ$-proximities defined on the same finite set; in contrast to metrics, $Σ$-proximities measure {\it comparative} proximity; the closer the objects, the greater the $Σ$-proximity; diagonal entries of the $Σ$-proximity matrix characterize the ``centrality'' of elements. The results are extended to arbitrary infinite sets. | |
| dc.description | 5 pages | |
| dc.identifier | https://arxiv.org/abs/math/0508183 | |
| dc.identifier | http://arxiv.org/abs/math/0508183 | |
| dc.identifier | Automation and Remote Control 59 (1998) 608--612 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/102167 | |
| dc.subject | Metric Geometry | |
| dc.subject | Data Structures and Algorithms | |
| dc.subject | Combinatorics | |
| dc.subject | 46F10; 54E40; 15A51 | |
| dc.title | On a Duality between Metrics and $Σ$-Proximities | |
| dc.type | text |