283:
522:
1364:
1032:
1449:, intersect pairwise and therefore by hyperconvexity have a common intersection; a ball of radius half the diameter centered at a point of this common intersection contains all of
711:
1275:
895:
447:
1775:
958:
842:
1296:
656:
629:
394:
1467:
1447:
1427:
1403:
1232:
1212:
1192:
1172:
1148:
1124:
1104:
1084:
1052:
915:
812:
792:
769:
731:
602:
582:
562:
542:
414:
367:
343:
323:
303:
199:
176:
152:
132:
92:
204:
1367:
1711:
1651:
1592:
54:. These properties can be defined in two seemingly different ways: hyperconvexity involves the intersection properties of
452:
1505:
748:
48:
1337:
963:
1823:
1501:
772:
661:
1587:
742:
1251:
1055:
1373:
Due to the equivalence between hyperconvexity and injectivity, these spaces are all also injective.
1489:
847:
1794:
103:
55:
40:
419:
1583:
1470:
931:
918:
1784:
1749:
1720:
1660:
1601:
1497:
1302:
821:
1806:
1761:
1732:
1698:
1674:
1631:
1615:
1281:
634:
607:
372:
1802:
1757:
1728:
1694:
1670:
1628:
1611:
20:
1706:
1493:
1477:
1452:
1432:
1412:
1388:
1217:
1197:
1177:
1157:
1133:
1109:
1089:
1069:
1037:
900:
797:
777:
754:
716:
587:
567:
547:
527:
399:
352:
328:
308:
288:
184:
161:
137:
117:
77:
1429:. This follows since the balls of radius half the diameter, centered at the points of
1817:
1753:
99:
1382:
107:
51:
32:
1588:"Extensions of uniformly continuous transformations and hyperconvex metric spaces"
158:
image of a line segment of length equal to the distance between the points (i.e.
1771:"Existence of fixed points of nonexpansive mappings in certain Banach lattices"
1740:
Sine, R. C. (1979). "On nonlinear contraction semigroups in sup norm spaces".
1481:
1323:
922:
1683:
1485:
1330:
36:
1665:
1639:
1606:
1406:
1278:
1151:
155:
59:
1798:
1724:
1312:) in the plane (which is equivalent up to rotation and scaling to the
1789:
1770:
1469:. Thus, injective spaces satisfy a particularly strong form of
66:
that these two different types of definitions are equivalent.
62:
of the space into larger spaces. However it is a theorem of
278:{\displaystyle {\bar {B}}_{r}(p)=\{q\mid d(p,q)\leq r\}}
1709:(1964). "Six theorems about injective metric spaces".
1537:
1455:
1435:
1415:
1391:
1340:
1284:
1254:
1220:
1200:
1180:
1160:
1136:
1112:
1092:
1072:
1040:
966:
934:
903:
850:
824:
800:
780:
757:
719:
664:
637:
610:
590:
570:
550:
530:
455:
422:
402:
375:
355:
331:
311:
291:
207:
187:
164:
140:
120:
80:
63:
1565:For additional properties of injective spaces see
1461:
1441:
1421:
1397:
1358:
1290:
1269:
1226:
1206:
1186:
1166:
1142:
1118:
1098:
1078:
1046:
1026:
952:
909:
889:
836:
806:
786:
763:
725:
705:
650:
623:
596:
576:
556:
536:
516:
441:
408:
388:
361:
337:
317:
297:
277:
193:
170:
146:
126:
86:
35:with certain properties generalizing those of the
1106:that is an image of a retraction. A metric space
1776:Proceedings of the American Mathematical Society
1566:
1242:Examples of hyperconvex metric spaces include
517:{\displaystyle r_{i}+r_{j}\geq d(p_{i},p_{j})}
1647:approach to some results on cuts and metrics"
58:in the space, while injectivity involves the
8:
272:
239:
1788:
1693:. Dordrecht: Kluwer Academic Publishers.
1664:
1605:
1454:
1434:
1414:
1390:
1381:In an injective space, the radius of the
1339:
1283:
1261:
1257:
1256:
1253:
1219:
1199:
1179:
1159:
1135:
1111:
1091:
1071:
1039:
965:
933:
902:
849:
823:
799:
779:
756:
718:
697:
675:
663:
642:
636:
615:
609:
589:
569:
549:
529:
505:
492:
473:
460:
454:
427:
421:
401:
380:
374:
369:is hyperconvex if, for any set of points
354:
330:
310:
290:
221:
210:
209:
206:
186:
163:
139:
119:
79:
1517:
1359:{\displaystyle \operatorname {Aim} (X)}
1027:{\displaystyle d(f(x),f(y))\leq d(x,y)}
1553:
1525:
1691:Handbook of Metric Fixed Point Theory
1488:) on a bounded injective space has a
7:
1684:"Introduction to hyperconvex spaces"
1682:Espínola, R.; Khamsi, M. A. (2001).
1549:
706:{\displaystyle d(p_{i},q)\leq r_{i}}
814:to a subspace of itself, such that
64:Aronszajn & Panitchpakdi (1956)
1689:. In Kirk, W. A.; Sims B. (eds.).
1486:nonexpansive mapping, or short map
1368:Metric space aimed at its subspace
14:
1712:Commentarii Mathematici Helvetici
1538:Aronszajn & Panitchpakdi 1956
305:meets, then there exists a point
1270:{\displaystyle \mathbb {R} ^{d}}
285:such that each pair of balls in
1652:Advances in Applied Mathematics
1319:), but not in higher dimensions
1593:Pacific Journal of Mathematics
1492:. A metric space is injective
1353:
1347:
1021:
1009:
1000:
997:
991:
982:
976:
970:
884:
878:
869:
866:
860:
854:
687:
668:
511:
485:
263:
251:
233:
227:
215:
201:is any family of closed balls
1:
1506:metric spaces and metric maps
349:Equivalently, a metric space
1754:10.1016/0362-546X(79)90055-5
890:{\displaystyle f(f(x))=f(x)}
1586:; Panitchpakdi, P. (1956).
1476:Every injective space is a
325:common to all the balls in
1840:
1567:Espínola & Khamsi 2001
740:
442:{\displaystyle r_{i}>0}
921:on its image (i.e. it is
953:{\displaystyle x,y\in X}
604:that is within distance
154:can be connected by the
29:hyperconvex metric space
1638:Chepoi, Victor (1997).
1666:10.1006/aama.1997.0549
1607:10.2140/pjm.1956.6.405
1463:
1443:
1423:
1399:
1385:that contains any set
1360:
1292:
1271:
1228:
1208:
1188:
1168:
1144:
1120:
1100:
1080:
1048:
1028:
954:
911:
891:
838:
837:{\displaystyle x\in X}
808:
788:
765:
727:
707:
652:
625:
598:
578:
558:
538:
518:
443:
410:
390:
363:
339:
319:
299:
279:
195:
172:
148:
128:
88:
25:injective metric space
1464:
1444:
1424:
1405:is equal to half the
1400:
1361:
1293:
1291:{\displaystyle \ell }
1272:
1229:
1209:
1189:
1169:
1145:
1121:
1101:
1081:
1049:
1029:
955:
912:
892:
839:
809:
789:
766:
743:Retraction (topology)
728:
708:
653:
651:{\displaystyle p_{i}}
626:
624:{\displaystyle r_{i}}
599:
579:
559:
539:
519:
444:
411:
391:
389:{\displaystyle p_{i}}
364:
340:
320:
300:
280:
196:
173:
149:
129:
89:
1453:
1433:
1413:
1389:
1338:
1282:
1252:
1218:
1198:
1178:
1158:
1134:
1110:
1090:
1070:
1038:
964:
932:
901:
848:
822:
798:
778:
755:
717:
662:
635:
608:
588:
568:
548:
528:
453:
420:
400:
373:
353:
329:
309:
289:
205:
185:
162:
138:
118:
78:
60:isometric embeddings
27:, or equivalently a
16:Type of metric space
1769:Soardi, P. (1979).
1620:Correction (1957),
1484:(or, equivalently,
564:, there is a point
1742:Nonlinear Analysis
1725:10.1007/BF02566944
1459:
1439:
1419:
1395:
1356:
1303:Manhattan distance
1288:
1267:
1224:
1204:
1184:
1164:
1140:
1116:
1096:
1076:
1044:
1024:
950:
907:
887:
834:
804:
784:
761:
751:of a metric space
723:
703:
648:
621:
594:
574:
554:
534:
514:
439:
406:
386:
359:
335:
315:
295:
275:
191:
168:
144:
124:
84:
1462:{\displaystyle S}
1442:{\displaystyle S}
1422:{\displaystyle S}
1398:{\displaystyle S}
1326:of a metric space
1227:{\displaystyle Y}
1207:{\displaystyle Z}
1187:{\displaystyle Y}
1167:{\displaystyle Z}
1143:{\displaystyle X}
1119:{\displaystyle X}
1099:{\displaystyle X}
1086:is a subspace of
1079:{\displaystyle X}
1047:{\displaystyle f}
919:identity function
910:{\displaystyle f}
807:{\displaystyle X}
787:{\displaystyle f}
764:{\displaystyle X}
726:{\displaystyle i}
597:{\displaystyle X}
577:{\displaystyle q}
557:{\displaystyle j}
537:{\displaystyle i}
409:{\displaystyle X}
362:{\displaystyle X}
338:{\displaystyle F}
318:{\displaystyle x}
298:{\displaystyle F}
218:
194:{\displaystyle F}
178:is a path space).
171:{\displaystyle X}
147:{\displaystyle y}
127:{\displaystyle x}
87:{\displaystyle X}
1831:
1810:
1792:
1765:
1736:
1702:
1688:
1678:
1668:
1622:Pacific J. Math.
1619:
1609:
1570:
1563:
1557:
1547:
1541:
1535:
1529:
1522:
1498:injective object
1468:
1466:
1465:
1460:
1448:
1446:
1445:
1440:
1428:
1426:
1425:
1420:
1404:
1402:
1401:
1396:
1365:
1363:
1362:
1357:
1297:
1295:
1294:
1289:
1276:
1274:
1273:
1268:
1266:
1265:
1260:
1233:
1231:
1230:
1225:
1214:is a retract of
1213:
1211:
1210:
1205:
1194:, that subspace
1193:
1191:
1190:
1185:
1173:
1171:
1170:
1165:
1149:
1147:
1146:
1141:
1125:
1123:
1122:
1117:
1105:
1103:
1102:
1097:
1085:
1083:
1082:
1077:
1053:
1051:
1050:
1045:
1033:
1031:
1030:
1025:
959:
957:
956:
951:
916:
914:
913:
908:
896:
894:
893:
888:
843:
841:
840:
835:
813:
811:
810:
805:
793:
791:
790:
785:
770:
768:
767:
762:
732:
730:
729:
724:
712:
710:
709:
704:
702:
701:
680:
679:
657:
655:
654:
649:
647:
646:
630:
628:
627:
622:
620:
619:
603:
601:
600:
595:
583:
581:
580:
575:
563:
561:
560:
555:
543:
541:
540:
535:
523:
521:
520:
515:
510:
509:
497:
496:
478:
477:
465:
464:
448:
446:
445:
440:
432:
431:
415:
413:
412:
407:
395:
393:
392:
387:
385:
384:
368:
366:
365:
360:
344:
342:
341:
336:
324:
322:
321:
316:
304:
302:
301:
296:
284:
282:
281:
276:
226:
225:
220:
219:
211:
200:
198:
197:
192:
177:
175:
174:
169:
153:
151:
150:
145:
133:
131:
130:
125:
106:have the binary
93:
91:
90:
85:
1839:
1838:
1834:
1833:
1832:
1830:
1829:
1828:
1814:
1813:
1790:10.2307/2042874
1768:
1739:
1705:
1686:
1681:
1645:
1637:
1582:
1579:
1574:
1573:
1564:
1560:
1548:
1544:
1536:
1532:
1523:
1519:
1514:
1451:
1450:
1431:
1430:
1411:
1410:
1387:
1386:
1379:
1336:
1335:
1318:
1311:
1280:
1279:
1255:
1250:
1249:
1240:
1216:
1215:
1196:
1195:
1176:
1175:
1156:
1155:
1132:
1131:
1108:
1107:
1088:
1087:
1068:
1067:
1036:
1035:
962:
961:
930:
929:
899:
898:
846:
845:
820:
819:
796:
795:
776:
775:
753:
752:
745:
739:
715:
714:
693:
671:
660:
659:
638:
633:
632:
611:
606:
605:
586:
585:
566:
565:
546:
545:
526:
525:
501:
488:
469:
456:
451:
450:
423:
418:
417:
398:
397:
376:
371:
370:
351:
350:
327:
326:
307:
306:
287:
286:
208:
203:
202:
183:
182:
160:
159:
136:
135:
116:
115:
114:Any two points
76:
75:
74:A metric space
72:
44:
21:metric geometry
17:
12:
11:
5:
1837:
1835:
1827:
1826:
1816:
1815:
1812:
1811:
1766:
1748:(6): 885–890.
1737:
1703:
1679:
1659:(4): 453–470.
1643:
1635:
1578:
1575:
1572:
1571:
1558:
1542:
1530:
1516:
1515:
1513:
1510:
1494:if and only if
1478:complete space
1471:Jung's theorem
1458:
1438:
1418:
1394:
1378:
1375:
1371:
1370:
1355:
1352:
1349:
1346:
1343:
1333:
1327:
1320:
1316:
1309:
1300:
1287:
1264:
1259:
1247:
1239:
1236:
1223:
1203:
1183:
1163:
1154:to a subspace
1139:
1126:is said to be
1115:
1095:
1075:
1060:
1059:
1043:
1023:
1020:
1017:
1014:
1011:
1008:
1005:
1002:
999:
996:
993:
990:
987:
984:
981:
978:
975:
972:
969:
949:
946:
943:
940:
937:
926:
906:
886:
883:
880:
877:
874:
871:
868:
865:
862:
859:
856:
853:
833:
830:
827:
803:
783:
760:
738:
735:
722:
700:
696:
692:
689:
686:
683:
678:
674:
670:
667:
645:
641:
618:
614:
593:
573:
553:
533:
513:
508:
504:
500:
495:
491:
487:
484:
481:
476:
472:
468:
463:
459:
438:
435:
430:
426:
405:
383:
379:
358:
347:
346:
334:
314:
294:
274:
271:
268:
265:
262:
259:
256:
253:
250:
247:
244:
241:
238:
235:
232:
229:
224:
217:
214:
190:
179:
167:
143:
123:
108:Helly property
94:is said to be
83:
71:
70:Hyperconvexity
68:
42:
15:
13:
10:
9:
6:
4:
3:
2:
1836:
1825:
1824:Metric spaces
1822:
1821:
1819:
1808:
1804:
1800:
1796:
1791:
1786:
1782:
1778:
1777:
1772:
1767:
1763:
1759:
1755:
1751:
1747:
1743:
1738:
1734:
1730:
1726:
1722:
1718:
1714:
1713:
1708:
1707:Isbell, J. R.
1704:
1700:
1696:
1692:
1685:
1680:
1676:
1672:
1667:
1662:
1658:
1654:
1653:
1648:
1646:
1636:
1633:
1630:
1626:
1623:
1617:
1613:
1608:
1603:
1599:
1595:
1594:
1589:
1585:
1584:Aronszajn, N.
1581:
1580:
1576:
1568:
1562:
1559:
1555:
1551:
1546:
1543:
1539:
1534:
1531:
1527:
1521:
1518:
1511:
1509:
1507:
1503:
1499:
1495:
1491:
1487:
1483:
1479:
1474:
1472:
1456:
1436:
1416:
1408:
1392:
1384:
1376:
1374:
1369:
1350:
1344:
1341:
1334:
1332:
1329:Any complete
1328:
1325:
1321:
1315:
1308:
1304:
1301:
1299:
1285:
1262:
1248:
1246:The real line
1245:
1244:
1243:
1237:
1235:
1221:
1201:
1181:
1161:
1153:
1137:
1130:if, whenever
1129:
1113:
1093:
1073:
1065:
1057:
1041:
1018:
1015:
1012:
1006:
1003:
994:
988:
985:
979:
973:
967:
960:we have that
947:
944:
941:
938:
935:
927:
924:
920:
904:
881:
875:
872:
863:
857:
851:
844:we have that
831:
828:
825:
817:
816:
815:
801:
781:
774:
758:
750:
744:
736:
734:
720:
698:
694:
690:
684:
681:
676:
672:
665:
643:
639:
616:
612:
591:
571:
551:
531:
506:
502:
498:
493:
489:
482:
479:
474:
470:
466:
461:
457:
436:
433:
428:
424:
403:
381:
377:
356:
332:
312:
292:
269:
266:
260:
257:
254:
248:
245:
242:
236:
230:
222:
212:
188:
180:
165:
157:
141:
121:
113:
112:
111:
109:
105:
101:
97:
81:
69:
67:
65:
61:
57:
53:
52:vector spaces
50:
46:
38:
34:
30:
26:
22:
1783:(1): 25–29.
1780:
1774:
1745:
1741:
1716:
1710:
1690:
1656:
1650:
1641:
1624:
1621:
1597:
1591:
1561:
1545:
1533:
1520:
1480:, and every
1475:
1383:minimum ball
1380:
1372:
1366:– see
1313:
1306:
1241:
1127:
1063:
1061:
1056:nonexpansive
746:
348:
104:closed balls
95:
73:
56:closed balls
33:metric space
28:
24:
18:
1600:: 405–439.
1554:Soardi 1979
1526:Chepoi 1997
1490:fixed point
1174:of a space
1066:of a space
1034:; that is,
897:; that is,
737:Injectivity
449:satisfying
110:. That is:
96:hyperconvex
49:dimensional
1577:References
1482:metric map
1377:Properties
1324:tight span
923:idempotent
749:retraction
741:See also:
658:(that is,
416:and radii
47:in higher-
1719:: 65–76.
1550:Sine 1979
1524:See e.g.
1496:it is an
1345:
1331:real tree
1286:ℓ
1277:with the
1152:isometric
1128:injective
1004:≤
945:∈
829:∈
691:≤
524:for each
480:≥
267:≤
246:∣
216:¯
156:isometric
98:if it is
45:distances
37:real line
1818:Category
1627:: 1729,
1502:category
1407:diameter
1298:distance
1238:Examples
928:for all
818:for all
794:mapping
773:function
713:for all
631:of each
102:and its
1807:0512051
1799:2042874
1762:0548959
1733:0182949
1699:1904284
1675:1479014
1632:0092146
1616:0084762
1500:in the
1064:retract
917:is the
39:and of
31:, is a
1805:
1797:
1760:
1731:
1697:
1673:
1614:
925:), and
100:convex
1795:JSTOR
1687:(PDF)
1512:Notes
771:is a
23:, an
1322:The
544:and
434:>
134:and
1785:doi
1750:doi
1721:doi
1661:doi
1640:"A
1602:doi
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1409:of
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733:).
584:in
396:in
181:If
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1598:6
1569:.
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240:{
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234:)
231:p
228:(
223:r
213:B
189:F
166:X
142:y
122:x
82:X
43:∞
41:L
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