20:
319:. However, as pointed out by Bernulf Weißbach, the first part of this claim is in fact false. But after improving a suboptimal conclusion within the corresponding derivation, one can indeed verify one of the constructed point sets as a counterexample for
1060:) so "it came as a surprise when Kahn and Kalai constructed finite sets showing the contrary". It's worth noting, however, that Karol Borsuk has formulated the problem just as a question, not suggesting the expected answer would be positive.
1020:
520:
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1800:
796:
438:
917:
852:
727:
698:
191:
140:
878:
943:
823:
1056:
As
Hinrichs and Richter say in the introduction to their work, the "Borsuk's conjecture believed by many to be true for some decades" (hence commonly called a
602:
582:
945:, that is if the gap between the volumes of the smallest and largest constant-width bodies grows exponentially. In 2024 a preprint by Arman, Bondarenko,
1825:
1952:
952:
1398:
455:
1031:
607:
353:. Shortly after, Thomas Jenrich derived a 64-dimensional counterexample from Bondarenko's construction, giving the best bound up to now.
1801:"Andrii Arman, Andriy Bondarenko, Fedor Nazarov, Andriy Prymak, and Danylo Radchenko Constructed Small Volume Bodies of Constant Width"
1188:
241:— shown by Julian Perkal (1947), and independently, 8 years later, by H. G. Eggleston (1955). A simple proof was found later by
1887:
1874:
Raigorodskii, Andreii M. (2008). "Three lectures on the Borsuk partition problem". In Young, Nicholas; Choi, Yemon (eds.).
1502:
1988:
1945:
1854:
1586:
1317:
1274:
1227:
732:
949:, Prymak, Radchenko reported to have answered this question in the affirmative giving a construction that satisfies
393:
2089:
1938:
1333:
1090:
883:
2063:
1272:(1946), "Mitteilung betreffend meine Note: Überdeckung einer Menge durch Mengen kleineren Durchmessers",
2094:
1998:
1879:
1863:
828:
703:
674:
167:
116:
2043:
1389:
1037:
287:
857:
1983:
1769:
Arman, Andrii; Bondarenko, Andriy; Nazarov, Fedor; Prymak, Andriy; Radchenko, Danylo (2024-05-28),
1139:
99:
2058:
2018:
2013:
2068:
922:
1451:
1167:
Perkal, Julian (1947), "Sur la subdivision des ensembles en parties de diamètre inférieur",
2053:
1683:
1648:
1609:
1563:
1535:
1429:
1368:
1295:
1248:
1209:
1186:
Eggleston, H. G. (1955), "Covering a three-dimensional set with sets of smaller diameter",
801:
329:
Their result was improved in 2003 by
Hinrichs and Richter, who constructed finite sets for
1897:
8:
2038:
2028:
279:
2033:
1687:
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2048:
2023:
1978:
1916:
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194:
63:
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242:
2008:
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1303:
1256:
88:
43:
1421:
1993:
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1417:
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1283:
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1197:
1148:
1103:
268:
1861:
Andrei M. Raigorodskii, The Borsuk partition problem: the seventieth anniversary,
346:
In 2013, Andriy V. Bondarenko had shown that Borsuk's conjecture is false for all
1644:
1605:
1581:
1531:
1425:
1364:
1291:
1244:
1205:
253:
67:
375:, mathematicians are interested in finding the general behavior of the function
2003:
1347:
Dekster, Boris (1995), "The Borsuk conjecture holds for fields of revolution",
16:
Can every bounded subset of Rn be partitioned into (n+1) smaller diameter sets?
1780:
1750:
1742:
Some old and new problems in combinatorial geometry I: Around Borsuk's problem
1701:
1640:
1526:
1201:
2083:
1725:
1320:[Borsuk's problem in three-dimensional spaces of constant curvature]
1269:
1222:
946:
257:
85:
1500:
Bondarenko, Andriy (2014) , "On Borsuk's
Conjecture for Two-Distance Sets",
1153:
1622:
1126:
1015:{\displaystyle {\text{Vol}}(K)\leq (0.9)^{n}{\text{Vol}}(\mathbb {B} ^{n})}
441:
59:
47:
1826:"Mathematicians Discover New Shapes to Solve Decades-Old Geometry Problem"
1627:
157:
1930:
1962:
1717:
1360:
1287:
1240:
1225:(1945), "Überdeckung einer Menge durch Mengen kleineren Durchmessers",
515:{\textstyle \alpha (n)\leq \left({\sqrt {3/2}}+\varepsilon \right)^{n}}
109:
Die folgende Frage bleibt offen: Lässt sich jede beschränkte
Teilmenge
38:
533:
is still unknown. However, it is conjectured that there is a constant
1921:
1678:
1661:
1486:
1412:
1393:
291:
1666:
Proceedings of the
International Congress of Mathematicians, Beijing
91:
of diameters smaller than the ball. At the same time he proved that
1770:
1740:
1550:
A 64-dimensional two-distance counterexample to Borsuk's conjecture
1099:
664:{\displaystyle {\text{Vol}}(K)=r^{n}{\text{Vol}}(\mathbb {B} ^{n})}
71:
70:
can be easily dissected into 4 solids, each of which has a smaller
1878:. London Mathematical Society Lecture Note Series. Vol. 347.
1600:
1584:(2014), "A 64-Dimensional Counterexample to Borsuk's Conjecture",
1558:
1516:
1133:[Three theorems about the n-dimensional Euclidean sphere]
825:
is the smallest effective radius of a body of constant width 2 in
222:
The question was answered in the positive in the following cases:
146:) Mengen zerlegen, von denen jede einen kleineren Durchmesser als
1318:"Проблема Борсука в трехмерных пространствах постоянной кривизны"
24:
19:
1911:
1478:
On the counterexamples to Borsuk's conjecture by Kahn and Kalai
95:
294:, who showed that the general answer to Borsuk's question is
1768:
1034:
on covering convex fields with smaller copies of themselves
1664:(2002), "Discrete mathematics: methods and challenges",
1131:"Drei Sätze über die n-dimensionale euklidische Sphäre"
564:
Oded
Schramm also worked in a related question, a body
1084:
Hinrichs, Aicke; Richter, Christian (28 August 2003).
458:
396:
232:— which is the original result by Karol Borsuk (1932).
955:
925:
886:
860:
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804:
735:
706:
677:
610:
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570:
215:
Drei Sätze über die n-dimensionale euklidische Sphäre
170:
119:
98:
are not enough in general. The proof is based on the
386:. Kahn and Kalai show that in general (that is, for
1396:(1993), "A counterexample to Borsuk's conjecture",
584:of constant width is said to have effective radius
1014:
937:
911:
872:
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576:
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432:
204:) sets, each of which has a smaller diameter than
185:
134:
2081:
440:many pieces. They also quote the upper bound by
298:. They claim that their construction shows that
1625:(1988), "Illuminating sets of constant width",
1083:
791:{\displaystyle {\sqrt {3+2/(n+1)}}-1\leq r_{n}}
326:(as well as all higher dimensions up to 1560).
156:The following question remains open: Can every
1579:
1946:
1702:"On the volume of sets having constant width"
1399:Bulletin of the American Mathematical Society
433:{\textstyle \alpha (n)\geq (1.2)^{\sqrt {n}}}
360:of dimensions such that the number of pieces
1873:
1050:
213:
107:
35:, for historical reasons incorrectly called
1079:
1077:
27:cut into three pieces of smaller diameter.
1953:
1939:
1499:
1189:Journal of the London Mathematical Society
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1119:
286:The problem was finally solved in 1993 by
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122:
102:. That led Borsuk to a general question:
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1315:
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18:
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1475:
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1116:
271:fields — shown by A.S. Riesling (1971).
2082:
1823:
1166:
1125:
356:Apart from finding the minimum number
78:-dimensional ball can be covered with
62:showed that an ordinary 3-dimensional
1934:
1912:
1798:
1772:Small volume bodies of constant width
1738:
1503:Discrete & Computational Geometry
912:{\displaystyle r_{n}\leq 1-\epsilon }
1660:
1086:"New sets with large Borsuk numbers"
522:. The correct order of magnitude of
1876:Surveys in contemporary mathematics
1587:Electronic Journal of Combinatorics
336:, which cannot be partitioned into
13:
1855:Algebraic Methods in Combinatorics
1846:
1459:Beiträge zur Algebra und Geometrie
14:
2106:
1905:
1275:Commentarii Mathematici Helvetici
1228:Commentarii Mathematici Helvetici
1332:, Kharkov State University (now
847:{\displaystyle \mathbb {R} ^{n}}
722:{\displaystyle \mathbb {R} ^{n}}
693:{\displaystyle \mathbb {B} ^{n}}
282:— shown by Boris Dekster (1995).
186:{\displaystyle \mathbb {R} ^{n}}
135:{\displaystyle \mathbb {R} ^{n}}
1817:
1792:
1762:
1732:
1693:
1654:
1615:
1573:
1541:
1493:
1469:
1452:"Sets with Large Borsuk Number"
1443:
1422:10.1090/S0273-0979-1993-00398-7
1382:
390:sufficiently large), one needs
1824:Barber, Gregory (2024-09-20).
1340:
1334:National University of Kharkiv
1309:
1262:
1215:
1179:
1160:
1009:
994:
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973:
967:
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873:{\displaystyle \epsilon >0}
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419:
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406:
400:
1:
1706:Israel Journal of Mathematics
1109:10.1016/S0012-365X(02)00833-6
1067:
74:than the ball, and generally
729:, he proved the lower bound
444:, who showed that for every
7:
1700:Schramm, Oded (June 1988).
1025:
343:parts of smaller diameter.
10:
2111:
1880:Cambridge University Press
1870:(2004), no. 3, 4–12.
1864:Mathematical Intelligencer
1450:Weißbach, Bernulf (2000),
854:and asked if there exists
305:pieces do not suffice for
53:
33:Borsuk problem in geometry
1969:
1799:Kalai, Gil (2024-05-31).
1781:10.48550/arXiv.2405.18501
1751:10.48550/arXiv.1505.04952
1739:Kalai, Gil (2015-05-19),
1641:10.1112/S0025579300015175
1527:10.1007/s00454-014-9579-4
256:convex fields — shown by
108:
1548:Jenrich, Thomas (2013),
1476:Jenrich, Thomas (2018),
1316:Riesling, A. S. (1971),
1202:10.1112/jlms/s1-30.1.11
1169:Colloquium Mathematicum
1154:10.4064/fm-20-1-177-190
1140:Fundamenta Mathematicae
1043:
938:{\displaystyle n\geq 2}
452:is sufficiently large,
1805:Combinatorics and more
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220:
214:
187:
136:
28:
2090:Disproved conjectures
1989:Euler's sum of powers
1917:"Borsuk's Conjecture"
1038:Kahn–Kalai conjecture
1032:Hadwiger's conjecture
1017:
940:
914:
875:
849:
820:
818:{\displaystyle r_{n}}
793:
724:
695:
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517:
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188:
137:
104:
22:
1882:. pp. 202–247.
1091:Discrete Mathematics
953:
923:
884:
858:
829:
802:
733:
704:
700:is the unit ball in
675:
608:
588:
568:
456:
394:
280:fields of revolution
168:
117:
46:. It is named after
1688:2002math.....12390A
1582:Brouwer, Andries E.
1568:2013arXiv1308.0206J
1349:Journal of Geometry
269:centrally-symmetric
100:Borsuk–Ulam theorem
42:, is a question in
1979:Chinese hypothesis
1914:Weisstein, Eric W.
1718:10.1007/BF02765037
1361:10.1007/BF01406827
1326:Ukr. Geom. Sbornik
1288:10.1007/BF02565947
1241:10.1007/BF02568103
1012:
935:
909:
870:
844:
815:
788:
719:
690:
661:
594:
574:
512:
430:
245:and Aladár Heppes.
183:
132:
29:
2095:Discrete geometry
2077:
2076:
1889:978-0-521-70564-6
1580:Jenrich, Thomas;
992:
959:
767:
641:
614:
597:{\displaystyle r}
577:{\displaystyle K}
493:
427:
44:discrete geometry
2102:
2029:Ono's inequality
1955:
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23:An example of a
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1965:
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1858:, course notes.
1852:Oleg Pikhurko,
1849:
1847:Further reading
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1843:
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1830:Quanta Magazine
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243:Branko Grünbaum
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68:Euclidean space
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2004:Hauptvermutung
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1906:External links
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1712:(2): 178–182.
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1510:(3): 509–515,
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1355:(1–2): 64–73,
1339:
1328:(in Russian),
1308:
1270:Hadwiger, Hugo
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1223:Hadwiger, Hugo
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1667:
1663:
1657:
1650:
1646:
1642:
1638:
1634:
1630:
1629:
1624:
1623:Schramm, Oded
1618:
1611:
1607:
1602:
1601:10.37236/4069
1597:
1594:(4): #P4.29,
1593:
1589:
1588:
1583:
1576:
1569:
1565:
1560:
1555:
1551:
1544:
1537:
1533:
1528:
1523:
1518:
1513:
1509:
1505:
1504:
1496:
1488:
1483:
1479:
1472:
1464:
1461:(in German),
1460:
1453:
1446:
1439:
1435:
1431:
1427:
1423:
1419:
1414:
1409:
1405:
1401:
1400:
1395:
1391:
1385:
1378:
1374:
1370:
1366:
1362:
1358:
1354:
1350:
1343:
1335:
1331:
1327:
1319:
1312:
1305:
1301:
1297:
1293:
1289:
1285:
1281:
1278:(in German),
1277:
1276:
1271:
1265:
1258:
1254:
1250:
1246:
1242:
1238:
1234:
1231:(in German),
1230:
1229:
1224:
1218:
1211:
1207:
1203:
1199:
1195:
1191:
1190:
1182:
1174:
1171:(in French),
1170:
1163:
1155:
1150:
1146:
1143:(in German),
1142:
1141:
1132:
1128:
1127:Borsuk, Karol
1122:
1120:
1110:
1105:
1101:
1097:
1093:
1092:
1087:
1080:
1078:
1073:
1059:
1053:
1049:
1039:
1036:
1033:
1030:
1029:
1023:
1004:
984:
976:
970:
964:
948:
932:
929:
926:
906:
903:
900:
897:
892:
888:
867:
864:
861:
839:
810:
806:
783:
779:
775:
772:
769:
761:
758:
755:
748:
744:
741:
738:
714:
685:
653:
633:
629:
625:
619:
591:
571:
562:
558:
552:
548:
544:
537:
530:
526:
507:
502:
498:
495:
490:
486:
482:
476:
471:
465:
459:
443:
424:
415:
409:
403:
397:
383:
379:
372:
368:
364:
354:
350:
344:
340:
333:
327:
323:
316:
312:and for each
309:
302:
295:
293:
289:
281:
273:
270:
262:
259:
258:Hugo Hadwiger
255:
247:
244:
238:
234:
229:
225:
224:
223:
216:
209:
201:
196:
178:
164:of the space
159:
154:
153:
127:
103:
101:
97:
90:
87:
82:
73:
69:
65:
61:
51:
49:
45:
41:
40:
34:
26:
21:
1999:Hedetniemi's
1973:
1920:
1875:
1867:
1862:
1853:
1833:. Retrieved
1829:
1819:
1808:. Retrieved
1804:
1794:
1784:, retrieved
1771:
1764:
1754:, retrieved
1741:
1734:
1709:
1705:
1695:
1679:math/0212390
1669:
1665:
1656:
1632:
1626:
1617:
1591:
1585:
1575:
1549:
1543:
1507:
1501:
1495:
1487:1809.09612v4
1477:
1471:
1465:(2): 417–423
1462:
1458:
1445:
1413:math/9307229
1406:(1): 60–62,
1403:
1397:
1384:
1352:
1348:
1342:
1329:
1325:
1311:
1282:(1): 72–73,
1279:
1273:
1264:
1235:(1): 73–75,
1232:
1226:
1217:
1193:
1187:
1181:
1172:
1168:
1162:
1144:
1138:
1095:
1089:
1057:
1052:
563:
556:
550:
546:
542:
535:
528:
524:
442:Oded Schramm
381:
377:
370:
366:
362:
355:
348:
345:
338:
331:
328:
321:
314:
307:
300:
285:
236:
227:
221:
199:
155:
106:
105:
80:
60:Karol Borsuk
57:
48:Karol Borsuk
36:
32:
30:
2059:Von Neumann
1963:conjectures
1672:: 119–135,
1628:Mathematika
1147:: 177–190,
1102:: 137–147.
195:partitioned
113:des Raumes
2084:Categories
2069:Williamson
2064:Weyl–Berry
2044:Schoen–Yau
1961:Disproved
1898:1144.52005
1835:2024-09-28
1810:2024-09-28
1786:2024-09-28
1756:2024-09-28
1662:Alon, Noga
1394:Kalai, Gil
1390:Kahn, Jeff
1068:References
1058:conjecture
880:such that
540:such that
39:conjecture
1922:MathWorld
1726:0021-2172
1559:1308.0206
1517:1305.2584
1438:119647518
1377:121586146
1304:121053805
1257:122199549
1196:: 11–24,
971:≤
930:≥
907:ϵ
904:−
898:≤
862:ϵ
776:≤
770:−
499:ε
472:≤
460:α
410:≥
398:α
317:> 2014
292:Gil Kalai
288:Jeff Kahn
58:In 1932,
37:Borsuk's
2039:Ragsdale
2019:Keller's
2014:Kalman's
1974:Borsuk's
1336:): 78–83
1129:(1933),
1100:Elsevier
1026:See also
919:for all
798:, where
671:, where
554:for all
274:For all
263:For all
248:For all
211:—
72:diameter
2049:Seifert
2024:Mertens
1684:Bibcode
1649:0986627
1610:3292266
1564:Bibcode
1536:3201240
1430:1193538
1369:1317256
1296:0017515
1249:0013901
1210:0067473
1098:(1–3).
947:Nazarov
549:) >
369:) >
260:(1946).
160:subset
158:bounded
96:subsets
86:compact
54:Problem
25:hexagon
2054:Tait's
2009:Hirsch
1984:Connes
1896:
1886:
1724:
1647:
1608:
1534:
1436:
1428:
1375:
1367:
1302:
1294:
1255:
1247:
1208:
538:> 1
324:= 1325
310:= 1325
254:smooth
197:into (
2034:Pólya
1994:Ganea
1674:arXiv
1554:arXiv
1512:arXiv
1482:arXiv
1455:(PDF)
1434:S2CID
1408:arXiv
1373:S2CID
1322:(PDF)
1300:S2CID
1253:S2CID
1135:(PDF)
448:, if
334:≥ 298
144:n + 1
1884:ISBN
1722:ISSN
1175:: 45
1044:Note
865:>
351:≥ 65
341:+ 11
290:and
278:for
267:for
252:for
150:hat?
142:in (
89:sets
64:ball
31:The
1894:Zbl
1777:doi
1747:doi
1714:doi
1637:doi
1596:doi
1522:doi
1418:doi
1357:doi
1284:doi
1237:doi
1198:doi
1149:doi
1104:doi
1096:270
991:Vol
977:0.9
958:Vol
640:Vol
613:Vol
604:if
559:≥ 1
416:1.2
373:+ 1
303:+ 1
239:= 3
230:= 2
202:+ 1
193:be
83:+ 1
66:in
2086::
1919:.
1892:.
1868:26
1828:.
1803:.
1775:,
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1480:,
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1457:,
1432:,
1426:MR
1424:,
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1404:29
1402:,
1392:;
1371:,
1365:MR
1363:,
1353:52
1351:,
1330:11
1324:,
1298:,
1292:MR
1290:,
1280:19
1251:,
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1204:,
1194:30
1192:,
1145:20
1137:,
1118:^
1094:.
1088:.
1076:^
1022:.
561:.
296:no
208:?
50:.
1954:e
1947:t
1940:v
1925:.
1900:.
1838:.
1813:.
1779::
1749::
1728:.
1716::
1686::
1676::
1670:1
1639::
1598::
1566::
1556::
1524::
1514::
1484::
1420::
1410::
1359::
1286::
1239::
1200::
1173:2
1151::
1112:.
1106::
1010:)
1005:n
1000:B
995:(
985:n
981:)
974:(
968:)
965:K
962:(
933:2
927:n
901:1
893:n
889:r
868:0
840:n
835:R
811:n
807:r
784:n
780:r
773:1
765:)
762:1
759:+
756:n
753:(
749:/
745:2
742:+
739:3
715:n
710:R
686:n
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659:)
654:n
649:B
644:(
634:n
630:r
626:=
623:)
620:K
617:(
592:r
572:K
557:n
551:c
547:n
545:(
543:α
536:c
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529:n
527:(
525:α
508:n
503:)
496:+
491:2
487:/
483:3
477:(
469:)
466:n
463:(
450:n
446:ε
425:n
420:)
413:(
407:)
404:n
401:(
388:n
384:)
382:n
380:(
378:α
371:n
367:n
365:(
363:α
358:n
349:n
339:n
332:n
322:n
315:n
308:n
301:n
276:n
265:n
250:n
237:n
228:n
206:E
200:n
179:n
174:R
162:E
148:E
128:n
123:R
111:E
93:n
81:n
76:n
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