50:
in the space. If two points are at a given distance from each other, it is natural to expect that one should be able to get from the first point to the second along a path whose arclength is equal to (or very close to) that distance. The distance between two points of a metric space relative to the
1407:
can be turned into a path metric space by defining the distance of two points as the infimum of the lengths of continuously differentiable curves connecting the two points. (The
Riemannian structure allows one to define the length of such curves.) Analogously, other manifolds in which a length is
729:
1285:
1204:
457:
586:
798:
1476:
666:
978:
316:
1573:
1347:
1244:
1600:
1507:
530:
495:
250:
1730:
223:
1148:
1113:
187:
1646:
1378:
1318:
893:
128:
1698:
1670:
1531:
1078:
1058:
1038:
1018:
998:
948:
921:
861:
841:
821:
626:
606:
407:
387:
360:
340:
270:
152:
678:
1719:
Herbert
Busemann, Selected Works, (Athanase Papadopoulos, ed.) Volume II, 842 p., Springer International Publishing, 2018.
1249:
1714:
Herbert
Busemann, Selected Works, (Athanase Papadopoulos, ed.) Volume I, 908 p., Springer International Publishing, 2018.
1764:
1739:
1156:
418:
1725:
543:
740:
66:
If the space has the stronger property that there always exists a path that achieves the infimum of length (a
1448:
957:
638:
279:
1539:
1323:
1220:
24:
154:
is a collection of points (such as all of the points in the plane, or all points on the circle) and
1782:
1578:
1485:
1435:. However, the converse does not hold, i.e. there exist length metric spaces that are not convex.
500:
465:
228:
1413:
28:
1613:
196:
1420:
1397:
20:
1118:
1083:
157:
1748:
1619:
1356:
1296:
866:
101:
87:
8:
1752:
1424:
1404:
1683:
1655:
1516:
1063:
1043:
1023:
1003:
983:
933:
906:
846:
826:
806:
611:
591:
392:
372:
345:
325:
255:
137:
1760:
1735:
1479:
1389:
537:
59:
of the lengths of all paths from the first point to the second. A metric space is a
1409:
19:"Geodesic distance" redirects here. For distances on the surface of a sphere, see
1649:
1215:
79:
47:
1606:
1510:
410:
1776:
131:
83:
39:
1605:
The metric of a length space has approximate midpoints. Conversely, every
1701:
1575:
is always a path metric space (with the caveat, as mentioned above, that
1432:
1291:
628:(this is consistent with the infimum definition since the infimum of the
35:
1677:
669:
63:
if the intrinsic metric agrees with the original metric of the space.
629:
43:
1673:
1396:
is not intrinsic, and the induced intrinsic metric is given by the
67:
319:
56:
1393:
1385:
1353:) is not a path metric space. The induced intrinsic metric on
1381:
90:
removed is not geodesic, but is still a length metric space.
1246:
with the ordinary
Euclidean metric is a path metric space.
1731:
Metric
Structures for Riemannian and Non-Riemannian Spaces
1609:
metric space with approximate midpoints is a length space.
27:. For the edge-count of a shortest path in a graph, see
1757:
1320:
with the metric inherited from the
Euclidean metric of
724:{\displaystyle (d_{\text{I}})_{\text{I}}=d_{\text{I}}.}
1388:, and the resulting length metric space is called the
641:
1686:
1658:
1622:
1581:
1542:
1519:
1488:
1451:
1359:
1326:
1299:
1252:
1223:
1159:
1121:
1086:
1066:
1046:
1026:
1006:
986:
960:
936:
909:
869:
849:
829:
809:
743:
681:
614:
594:
546:
503:
468:
421:
395:
375:
348:
328:
282:
258:
231:
199:
160:
140:
104:
1280:{\displaystyle \mathbb {R} ^{n}\smallsetminus \{0\}}
1692:
1664:
1640:
1594:
1567:
1525:
1501:
1470:
1372:
1341:
1312:
1279:
1238:
1198:
1142:
1107:
1072:
1052:
1032:
1012:
992:
972:
942:
915:
887:
855:
835:
815:
792:
723:
660:
620:
600:
580:
524:
489:
451:
401:
381:
354:
334:
310:
264:
244:
217:
181:
146:
122:
23:. For distances on the surface of the Earth, see
1774:
1734:, Progress in Math., vol. 152, Birkhäuser,
1392:. In two dimensions, the chordal metric on the
86:as its geodesics. The Euclidean plane with the
1199:{\displaystyle {d(x,y) \over 2}+\varepsilon .}
1274:
1268:
536:of such a path is defined as explained for
452:{\displaystyle \gamma \colon \rightarrow M}
1747:
1428:
588:if there is no path of finite length from
1329:
1255:
1226:
632:within the closed interval is +∞).
581:{\displaystyle d_{\text{I}}(x,y)=\infty }
793:{\displaystyle d_{\text{I}}(x,y)=d(x,y)}
189:is a function that provides us with the
1775:
1724:
1513:than or equal to the one defined by
13:
1471:{\displaystyle d\leq d_{\text{I}}}
661:{\textstyle d\mapsto d_{\text{I}}}
575:
14:
1794:
973:{\displaystyle \varepsilon >0}
322:of the lengths of all paths from
311:{\displaystyle d_{\text{I}}(x,y)}
1568:{\displaystyle (M,d_{\text{I}})}
1342:{\displaystyle \mathbb {R} ^{2}}
1239:{\displaystyle \mathbb {R} ^{n}}
1635:
1623:
1616:states that if a length space
1562:
1543:
1178:
1166:
1137:
1125:
1102:
1090:
882:
870:
787:
775:
766:
754:
696:
682:
645:
569:
557:
513:
507:
478:
472:
443:
440:
428:
305:
293:
176:
164:
117:
105:
93:
1:
1708:
1439:
1431:, Theorem 2.16), a result of
1595:{\displaystyle d_{\text{I}}}
1502:{\displaystyle d_{\text{I}}}
525:{\displaystyle \gamma (1)=y}
490:{\displaystyle \gamma (0)=x}
245:{\displaystyle d_{\text{I}}}
16:Concept in geometry/topology
7:
1209:
10:
1799:
1427:is a length metric space (
82:is a geodesic space, with
18:
225:. We define a new metric
25:Geodesics on an ellipsoid
1414:sub-Riemannian manifolds
980:and any pair of points
274:induced intrinsic metric
218:{\displaystyle x,y\in M}
1652:then any two points in
1150:are both smaller than
930:We say that the metric
42:, one can consider the
29:Distance (graph theory)
1694:
1672:can be connected by a
1666:
1642:
1596:
1569:
1527:
1503:
1472:
1429:Khamsi & Kirk 2001
1380:measures distances as
1374:
1343:
1314:
1281:
1240:
1200:
1144:
1143:{\displaystyle d(c,y)}
1109:
1108:{\displaystyle d(x,c)}
1074:
1054:
1034:
1014:
994:
974:
944:
917:
889:
857:
837:
817:
794:
725:
662:
622:
602:
582:
526:
491:
453:
403:
383:
356:
336:
312:
266:
246:
219:
183:
182:{\displaystyle d(x,y)}
148:
124:
70:) then it is called a
1695:
1667:
1643:
1641:{\displaystyle (M,d)}
1597:
1570:
1528:
1504:
1473:
1398:great-circle distance
1375:
1373:{\displaystyle S^{1}}
1344:
1315:
1313:{\displaystyle S^{1}}
1282:
1241:
1201:
1145:
1110:
1075:
1055:
1035:
1015:
995:
975:
952:approximate midpoints
945:
918:
890:
888:{\displaystyle (M,d)}
858:
838:
818:
795:
726:
663:
623:
603:
583:
527:
492:
454:
404:
384:
357:
337:
313:
267:
247:
220:
184:
149:
125:
123:{\displaystyle (M,d)}
72:geodesic metric space
21:Great-circle distance
1684:
1656:
1620:
1579:
1540:
1517:
1509:is therefore always
1486:
1449:
1445:In general, we have
1357:
1324:
1297:
1250:
1221:
1157:
1119:
1084:
1064:
1044:
1024:
1004:
984:
958:
934:
907:
867:
847:
827:
807:
741:
679:
639:
612:
592:
544:
501:
466:
419:
393:
373:
346:
326:
280:
256:
229:
197:
158:
138:
102:
78:. For instance, the
1674:minimizing geodesic
1425:convex metric space
1405:Riemannian manifold
61:length metric space
1749:Khamsi, Mohamed A.
1690:
1662:
1638:
1614:Hopf–Rinow theorem
1592:
1565:
1523:
1499:
1468:
1370:
1339:
1310:
1277:
1236:
1196:
1140:
1105:
1070:
1050:
1030:
1010:
990:
970:
940:
913:
885:
853:
833:
813:
790:
721:
658:
618:
598:
578:
538:rectifiable curves
522:
487:
449:
399:
379:
352:
332:
308:
262:
242:
215:
179:
144:
120:
55:is defined as the
1693:{\displaystyle M}
1665:{\displaystyle M}
1602:can be infinite).
1589:
1559:
1526:{\displaystyle d}
1496:
1465:
1410:Finsler manifolds
1408:defined included
1390:Riemannian circle
1185:
1073:{\displaystyle M}
1053:{\displaystyle c}
1033:{\displaystyle M}
1013:{\displaystyle y}
993:{\displaystyle x}
943:{\displaystyle d}
916:{\displaystyle d}
901:path metric space
856:{\displaystyle M}
836:{\displaystyle y}
816:{\displaystyle x}
751:
715:
702:
692:
655:
621:{\displaystyle y}
601:{\displaystyle x}
554:
402:{\displaystyle y}
382:{\displaystyle x}
355:{\displaystyle y}
335:{\displaystyle x}
290:
265:{\displaystyle M}
239:
147:{\displaystyle M}
1790:
1769:
1753:Kirk, William A.
1744:
1699:
1697:
1696:
1691:
1676:and all bounded
1671:
1669:
1668:
1663:
1648:is complete and
1647:
1645:
1644:
1639:
1601:
1599:
1598:
1593:
1591:
1590:
1587:
1574:
1572:
1571:
1566:
1561:
1560:
1557:
1532:
1530:
1529:
1524:
1508:
1506:
1505:
1500:
1498:
1497:
1494:
1477:
1475:
1474:
1469:
1467:
1466:
1463:
1403:Every connected
1379:
1377:
1376:
1371:
1369:
1368:
1348:
1346:
1345:
1340:
1338:
1337:
1332:
1319:
1317:
1316:
1311:
1309:
1308:
1286:
1284:
1283:
1278:
1264:
1263:
1258:
1245:
1243:
1242:
1237:
1235:
1234:
1229:
1205:
1203:
1202:
1197:
1186:
1181:
1161:
1149:
1147:
1146:
1141:
1114:
1112:
1111:
1106:
1079:
1077:
1076:
1071:
1059:
1057:
1056:
1051:
1039:
1037:
1036:
1031:
1019:
1017:
1016:
1011:
999:
997:
996:
991:
979:
977:
976:
971:
949:
947:
946:
941:
922:
920:
919:
914:
894:
892:
891:
886:
862:
860:
859:
854:
842:
840:
839:
834:
822:
820:
819:
814:
799:
797:
796:
791:
753:
752:
749:
730:
728:
727:
722:
717:
716:
713:
704:
703:
700:
694:
693:
690:
667:
665:
664:
659:
657:
656:
653:
627:
625:
624:
619:
607:
605:
604:
599:
587:
585:
584:
579:
556:
555:
552:
531:
529:
528:
523:
496:
494:
493:
488:
458:
456:
455:
450:
408:
406:
405:
400:
388:
386:
385:
380:
361:
359:
358:
353:
341:
339:
338:
333:
317:
315:
314:
309:
292:
291:
288:
271:
269:
268:
263:
251:
249:
248:
243:
241:
240:
237:
224:
222:
221:
216:
188:
186:
185:
180:
153:
151:
150:
145:
129:
127:
126:
121:
53:intrinsic metric
1798:
1797:
1793:
1792:
1791:
1789:
1788:
1787:
1783:Metric geometry
1773:
1772:
1767:
1742:
1726:Gromov, Mikhail
1711:
1685:
1682:
1681:
1657:
1654:
1653:
1650:locally compact
1621:
1618:
1617:
1586:
1582:
1580:
1577:
1576:
1556:
1552:
1541:
1538:
1537:
1518:
1515:
1514:
1493:
1489:
1487:
1484:
1483:
1462:
1458:
1450:
1447:
1446:
1442:
1364:
1360:
1358:
1355:
1354:
1333:
1328:
1327:
1325:
1322:
1321:
1304:
1300:
1298:
1295:
1294:
1259:
1254:
1253:
1251:
1248:
1247:
1230:
1225:
1224:
1222:
1219:
1218:
1216:Euclidean space
1212:
1162:
1160:
1158:
1155:
1154:
1120:
1117:
1116:
1085:
1082:
1081:
1065:
1062:
1061:
1045:
1042:
1041:
1025:
1022:
1021:
1005:
1002:
1001:
985:
982:
981:
959:
956:
955:
935:
932:
931:
908:
905:
904:
903:and the metric
868:
865:
864:
848:
845:
844:
828:
825:
824:
808:
805:
804:
803:for all points
748:
744:
742:
739:
738:
712:
708:
699:
695:
689:
685:
680:
677:
676:
652:
648:
640:
637:
636:
613:
610:
609:
593:
590:
589:
551:
547:
545:
542:
541:
502:
499:
498:
467:
464:
463:
420:
417:
416:
394:
391:
390:
374:
371:
370:
347:
344:
343:
327:
324:
323:
287:
283:
281:
278:
277:
272:, known as the
257:
254:
253:
236:
232:
230:
227:
226:
198:
195:
194:
193:between points
159:
156:
155:
139:
136:
135:
103:
100:
99:
96:
80:Euclidean plane
32:
17:
12:
11:
5:
1796:
1786:
1785:
1771:
1770:
1765:
1759:, Wiley-IEEE,
1745:
1740:
1721:
1720:
1716:
1715:
1710:
1707:
1706:
1705:
1689:
1661:
1637:
1634:
1631:
1628:
1625:
1610:
1603:
1585:
1564:
1555:
1551:
1548:
1545:
1534:
1522:
1492:
1461:
1457:
1454:
1441:
1438:
1437:
1436:
1417:
1401:
1367:
1363:
1351:chordal metric
1336:
1331:
1307:
1303:
1288:
1276:
1273:
1270:
1267:
1262:
1257:
1233:
1228:
1211:
1208:
1207:
1206:
1195:
1192:
1189:
1184:
1180:
1177:
1174:
1171:
1168:
1165:
1139:
1136:
1133:
1130:
1127:
1124:
1104:
1101:
1098:
1095:
1092:
1089:
1069:
1049:
1029:
1009:
989:
969:
966:
963:
939:
912:
884:
881:
878:
875:
872:
863:, we say that
852:
832:
812:
801:
800:
789:
786:
783:
780:
777:
774:
771:
768:
765:
762:
759:
756:
747:
732:
731:
720:
711:
707:
698:
688:
684:
651:
647:
644:
617:
597:
577:
574:
571:
568:
565:
562:
559:
550:
521:
518:
515:
512:
509:
506:
486:
483:
480:
477:
474:
471:
460:
459:
448:
445:
442:
439:
436:
433:
430:
427:
424:
411:continuous map
398:
378:
351:
331:
307:
304:
301:
298:
295:
286:
276:, as follows:
261:
235:
214:
211:
208:
205:
202:
178:
175:
172:
169:
166:
163:
143:
119:
116:
113:
110:
107:
95:
92:
76:geodesic space
15:
9:
6:
4:
3:
2:
1795:
1784:
1781:
1780:
1778:
1768:
1766:0-471-41825-0
1762:
1758:
1754:
1750:
1746:
1743:
1741:0-8176-3898-9
1737:
1733:
1732:
1727:
1723:
1722:
1718:
1717:
1713:
1712:
1703:
1687:
1679:
1675:
1659:
1651:
1632:
1629:
1626:
1615:
1611:
1608:
1604:
1583:
1553:
1549:
1546:
1535:
1520:
1512:
1490:
1481:
1459:
1455:
1452:
1444:
1443:
1434:
1430:
1426:
1422:
1418:
1415:
1411:
1406:
1402:
1399:
1395:
1391:
1387:
1383:
1365:
1361:
1352:
1334:
1305:
1301:
1293:
1289:
1271:
1265:
1260:
1231:
1217:
1214:
1213:
1193:
1190:
1187:
1182:
1175:
1172:
1169:
1163:
1153:
1152:
1151:
1134:
1131:
1128:
1122:
1099:
1096:
1093:
1087:
1067:
1047:
1040:there exists
1027:
1007:
987:
967:
964:
961:
953:
937:
928:
926:
910:
902:
898:
879:
876:
873:
850:
830:
810:
784:
781:
778:
772:
769:
763:
760:
757:
745:
737:
736:
735:
718:
709:
705:
686:
675:
674:
673:
671:
649:
642:
633:
631:
615:
595:
572:
566:
563:
560:
548:
539:
535:
519:
516:
510:
504:
484:
481:
475:
469:
446:
437:
434:
431:
425:
422:
415:
414:
413:
412:
396:
376:
368:
363:
349:
329:
321:
302:
299:
296:
284:
275:
259:
233:
212:
209:
206:
203:
200:
192:
173:
170:
167:
161:
141:
133:
114:
111:
108:
91:
89:
85:
84:line segments
81:
77:
73:
69:
64:
62:
58:
54:
49:
45:
41:
40:metric spaces
37:
30:
26:
22:
1756:
1729:
1350:
951:
929:
924:
900:
897:length space
896:
802:
733:
635:The mapping
634:
533:
461:
366:
364:
273:
190:
132:metric space
97:
75:
71:
65:
60:
52:
36:mathematical
33:
1678:closed sets
1482:defined by
1433:Karl Menger
1292:unit circle
1287:is as well.
954:if for any
94:Definitions
1709:References
1536:The space
1440:Properties
1080:such that
670:idempotent
1456:≤
1266:∖
1191:ε
962:ε
925:intrinsic
646:↦
630:empty set
576:∞
540:. We set
505:γ
470:γ
444:→
426::
423:γ
210:∈
44:arclength
38:study of
1777:Category
1755:(2001),
1728:(1999),
1607:complete
1480:topology
1478:and the
1421:complete
1210:Examples
365:Here, a
191:distance
134:, i.e.,
68:geodesic
1702:compact
1386:radians
672:, i.e.
320:infimum
318:is the
57:infimum
34:In the
1763:
1738:
1394:sphere
1382:angles
534:length
532:. The
88:origin
1511:finer
1349:(the
899:or a
895:is a
462:with
409:is a
369:from
130:be a
48:paths
1761:ISBN
1736:ISBN
1700:are
1612:The
1423:and
1419:Any
1412:and
1290:The
1115:and
1000:and
965:>
950:has
823:and
497:and
367:path
98:Let
1680:in
1384:in
1060:in
1020:in
923:is
843:in
734:If
668:is
608:to
389:to
342:to
252:on
74:or
46:of
1779::
1751:;
927:.
362:.
1704:.
1688:M
1660:M
1636:)
1633:d
1630:,
1627:M
1624:(
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1584:d
1563:)
1558:I
1554:d
1550:,
1547:M
1544:(
1533:.
1521:d
1495:I
1491:d
1464:I
1460:d
1453:d
1416:.
1400:.
1366:1
1362:S
1335:2
1330:R
1306:1
1302:S
1275:}
1272:0
1269:{
1261:n
1256:R
1232:n
1227:R
1194:.
1188:+
1183:2
1179:)
1176:y
1173:,
1170:x
1167:(
1164:d
1138:)
1135:y
1132:,
1129:c
1126:(
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1103:)
1100:c
1097:,
1094:x
1091:(
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1008:y
988:x
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883:)
880:d
877:,
874:M
871:(
851:M
831:y
811:x
788:)
785:y
782:,
779:x
776:(
773:d
770:=
767:)
764:y
761:,
758:x
755:(
750:I
746:d
719:.
714:I
710:d
706:=
701:I
697:)
691:I
687:d
683:(
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650:d
643:d
616:y
596:x
573:=
570:)
567:y
564:,
561:x
558:(
553:I
549:d
520:y
517:=
514:)
511:1
508:(
485:x
482:=
479:)
476:0
473:(
447:M
441:]
438:1
435:,
432:0
429:[
397:y
377:x
350:y
330:x
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303:y
300:,
297:x
294:(
289:I
285:d
260:M
238:I
234:d
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204:,
201:x
177:)
174:y
171:,
168:x
165:(
162:d
142:M
118:)
115:d
112:,
109:M
106:(
31:.
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