31:
52:
1000:
691:
2632:
995:{\displaystyle {\begin{aligned}V_{2k}(r)&={\frac {\pi ^{k}}{k!}}r^{2k}\,,\\V_{2k+1}(r)&={\frac {2^{k+1}\pi ^{k}}{\left(2k+1\right)!!}}r^{2k+1}={\frac {2\left(k!\right)\left(4\pi \right)^{k}}{\left(2k+1\right)!}}r^{2k+1}\,.\end{aligned}}}
2400:
666:
2905:
2425:
1622:
696:
1237:
1350:
1696:
1777:
2246:
569:
436:
403:
1490:
2820:
2668:
1944:
1888:
2815:
2122:
1980:
1530:
2241:
1848:
1819:
2180:
350:
323:
2148:
2420:
2080:
2060:
2040:
2020:
2000:
1716:
1095:
370:
296:
2954:. Note this theorem does not hold if "open" subset is replaced by "closed" subset, because the origin point qualifies but does not define a norm on
688:
at the integers and half integers gives formulas for the volume of a
Euclidean ball that do not require an evaluation of the gamma function. These are:
2627:{\displaystyle B(r)=\left\{x\in \mathbb {R} ^{n}\,:\left\|x\right\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\dots +|x_{n}|^{p}\right)^{1/p}<r\right\}.}
685:
1535:
1158:
1271:
69:
3282:
135:
3268:
116:
88:
73:
3395:
3346:
95:
3192:
1631:
1725:
102:
62:
3083:
3438:
408:
375:
2719:-norm, known as the Euclidean metric, generates the well known disks within circles, and for other values of
84:
3217:
3098:
1450:
3004:
498:
1375:
252:
2644:
3365:
3315:
1893:
1068:
3393:
Gruber, Peter M. (1982). "Isometries of the space of convex bodies contained in a
Euclidean ball".
3227:
2395:{\displaystyle \left\|x\right\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\dots +|x_{n}|^{p}\right)^{1/p},}
1858:
1399:
545:
661:{\displaystyle V_{n}(r)={\frac {\pi ^{\frac {n}{2}}}{\Gamma \left({\frac {n}{2}}+1\right)}}r^{n},}
3412:
3381:
3355:
3332:
3181:
2941:
2703:
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1379:
510:
263:
232:
1867:
2794:
2088:
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3278:
3144:
2969:
2900:{\displaystyle \lVert x\rVert _{\infty }=\max\{\left|x_{1}\right|,\dots ,\left|x_{n}\right|\}}
1949:
1499:
1429:
109:
2220:
1824:
1782:
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3404:
3373:
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3136:
2915:
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1387:
1005:
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328:
301:
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1719:
1425:
1414:
240:
228:
186:
175:
35:
2127:
3369:
2184:
The
Euclidean balls discussed earlier are an example of balls in a normed vector space.
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2778:
2405:
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2025:
2005:
1985:
1701:
1391:
1080:
677:
673:
355:
281:
157:
3427:
3416:
3385:
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3164:, bounded by two planes passing through a sphere center and the surface of the sphere
3128:
3102:
2724:
1395:
1417:
if, given any positive radius, it is covered by finitely many balls of that radius.
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1421:
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256:
190:
30:
3186:
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1410:
494:
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51:
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can be classified in two classes, that can be identified with the two possible
3236:
2927:
1403:
1260:, is likewise defined as the set of points of distance less than or equal to
3274:
3025:
1383:
1367:
681:
1374:
A ball in a general metric space need not be round. For example, a ball in
17:
3222:
3208:
3203:
2923:
275:
248:
217:
1617:{\displaystyle B_{r}(p)\subseteq {\overline {B_{r}(p)}}\subseteq B_{r},}
3408:
3360:
3336:
3313:
Smith, D. J.; Vamanamurthy, M. K. (1989). "How small is a unit ball?".
2950:
where the balls are all translated and uniformly scaled copies of
1143:, is defined the same way as a Euclidean ball, as the set of points in
3197:
3140:, bounded by a conical boundary with apex at the center of the sphere
3118:
2205:
526:
515:
244:
236:
162:
1402:
is never compact. However, a ball in a vector space will always be
29:
3344:
Dowker, J. S. (1996). "Robin
Conditions on the Euclidean ball".
3175:
684:
function to fractional arguments). Using explicit formulas for
45:
1436:
of open balls. This topology on a metric space is called the
2975:, not necessarily induced by a metric. An (open or closed)
478:
is the set of all points of distance less than or equal to
2710:
parallel to the coordinate axes as their boundaries. The
1398:. For example, a closed ball in any infinite-dimensional
185:
These concepts are defined not only in three-dimensional
3302:
2693:
parallel to the coordinate axes; those according to the
3123:
A number of special regions can be defined for a ball:
1354:
In particular, a ball (open or closed) always includes
3156:, bounded by two concentric spheres of differing radii
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2008:
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1232:{\displaystyle B_{r}(p)=\{x\in M\mid d(x,p)<r\}.}
1532:in this topology. While it is always the case that
76:. Unsourced material may be challenged and removed.
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2014:
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1345:{\displaystyle B_{r}=\{x\in M\mid d(x,p)\leq r\}.}
1344:
1231:
1089:
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660:
554:-dimensional volume of a Euclidean ball of radius
430:
397:
364:
344:
317:
290:
189:but also for lower and higher dimensions, and for
2843:
2402:Then an open ball around the origin with radius
1004:In the formula for odd-dimensional volumes, the
680:(which can be thought of as an extension of the
2746:- balls are within octahedra with axes-aligned
2723:, the corresponding balls are areas bounded by
462:is the set of all points of distance less than
3299:NIST Digital Library of Mathematical Functions
3018:-ball is homeomorphic to the Cartesian space
1691:{\displaystyle {\overline {B_{r}(p)}}=B_{r}.}
1406:as a consequence of the triangle inequality.
1099:be a positive real number. The open (metric)
8:
2894:
2846:
2831:
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1930:
1918:
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1772:{\displaystyle {\overline {B_{1}(p)}}=\{p\}}
1766:
1760:
1336:
1297:
1223:
1184:
2759:-balls are within cubes with axes-aligned
2689:metric) are bounded by squares with their
1432:, the open sets of which are all possible
1413:if it is contained in some ball. A set is
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431:{\displaystyle \operatorname {int} D^{n}}
422:
410:
398:{\displaystyle \operatorname {int} B^{n}}
389:
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336:
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283:
136:Learn how and when to remove this message
1371:(open or closed) is a ball of radius 1.
3259:
1890:is also a metric space with the metric
686:particular values of the gamma function
3270:Encyclopedic Dictionary of Mathematics
3148:, bounded by a pair of parallel planes
2788:generates the inner of usual spheres.
1485:{\displaystyle {\overline {B_{r}(p)}}}
1358:itself, since the definition requires
298:-dimensional ball is often denoted as
3068:. The homeomorphisms between an open
7:
3043:-ball is homeomorphic to the closed
2791:Often can also consider the case of
74:adding citations to reliable sources
493:-space, every ball is bounded by a
227:. Thus, for example, a ball in the
3101:; if it is smooth, it need not be
2835:
2804:
2763:, and the boundaries of balls for
1946:In such spaces, an arbitrary ball
613:
178:that constitute the sphere) or an
25:
2995:to an (open or closed) Euclidean
2727:(hypoellipses or hyperellipses).
1394:. A closed ball also need not be
1245:(metric) ball, sometimes denoted
566:-dimensional Euclidean space is:
42:is the volume bounded by a sphere
3189:, an extension to negative radii
2968:One may talk about balls in any
2706:metric, have squares with their
2663:{\displaystyle \mathbb {R} ^{2}}
50:
27:Volume space bounded by a sphere
1939:{\displaystyle d(x,y)=\|x-y\|.}
1698:For example, in a metric space
61:needs additional citations for
3329:10.1080/0025570x.1989.11977419
2581:
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2545:
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2042:may be viewed as a scaled (by
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1409:A subset of a metric space is
1327:
1315:
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1285:
1214:
1202:
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1172:
796:
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718:
712:
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262:In other contexts, such as in
1:
3396:Israel Journal of Mathematics
3347:Classical and Quantum Gravity
2022:with a distance of less than
3304:Release 1.0.6 of 2013-05-06.
3054:-ball is homeomorphic to an
3007:, as the building blocks of
1752:
1658:
1584:
1477:
1015:is defined for odd integers
243:, a ball is taken to be the
3200:, a similar geometric shape
3193:Neighbourhood (mathematics)
2670:, "balls" according to the
2641:, in a 2-dimensional plane
2217:, that is one chooses some
2124:Such "centered" balls with
1029:+ 1)!! = 1 ⋅ 3 ⋅ 5 ⋅ ⋯ ⋅ (2
3465:
3378:10.1088/0264-9381/13/4/003
3116:
2914:More generally, given any
1883:{\displaystyle \|\cdot \|}
543:
270:is sometimes used to mean
3039:. Any closed topological
2810:{\displaystyle p=\infty }
2117:{\displaystyle B_{1}(0).}
497:. The ball is a bounded
3267:Sūgakkai, Nihon (1993).
3084:topological orientations
3003:-balls are important in
2817:in which case we define
2679:-norm (often called the
1975:{\displaystyle B_{r}(y)}
1525:{\displaystyle B_{r}(p)}
1043:In general metric spaces
235:, the area bounded by a
3218:Alexander horned sphere
2702:-norm, also called the
2236:{\displaystyle p\geq 1}
1853:In normed vector spaces
1843:{\displaystyle p\in X.}
1814:{\displaystyle B_{1}=X}
1386:, and a ball under the
231:is the same thing as a
201:dimensions is called a
85:"Ball" mathematics
3132:, bounded by one plane
3005:combinatorial topology
2901:
2811:
2664:
2628:
2416:
2396:
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2176:
2144:
2118:
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2016:
1996:
1976:
1940:
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1844:
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1526:
1486:
1428:, giving this space a
1346:
1233:
1147:of distance less than
1091:
996:
662:
525:, and is bounded by a
432:
399:
366:
346:
319:
292:
166:; it is also called a
43:
3058:-ball if and only if
3014:Any open topological
2964:In topological spaces
2902:
2812:
2665:
2629:
2417:
2397:
2238:
2177:
2175:{\displaystyle B(r).}
2145:
2119:
2077:
2062:) and translated (by
2057:
2037:
2017:
1997:
1977:
1941:
1885:
1845:
1816:
1774:
1713:
1693:
1628:always the case that
1619:
1527:
1487:
1376:real coordinate space
1347:
1234:
1114:, usually denoted by
1092:
997:
663:
433:
400:
372:-dimensional ball is
367:
347:
345:{\displaystyle D^{n}}
320:
318:{\displaystyle B^{n}}
293:
253:one-dimensional space
33:
3316:Mathematics Magazine
2821:
2795:
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2422:is given by the set
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2006:
1986:
1950:
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1702:
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1420:The open balls of a
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1159:
1106:centered at a point
1081:
1071:(distance function)
692:
570:
409:
376:
356:
329:
302:
282:
249:2-dimensional sphere
212:and is bounded by a
70:improve this article
3370:1996CQGra..13..585D
2999:-ball. Topological
2940:, one can define a
2916:centrally symmetric
2910:General convex norm
2143:{\displaystyle y=0}
1859:normed vector space
1438:topology induced by
1400:normed vector space
546:Volume of an n-ball
3409:10.1007/BF02761407
3239:– a 3-ball in the
3182:Disk (mathematics)
3178:– ordinary meaning
3097:-ball need not be
2897:
2807:
2660:
2624:
2412:
2392:
2233:
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2140:
2114:
2072:
2052:
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2012:
1992:
1972:
1936:
1880:
1840:
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1708:
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1482:
1380:Chebyshev distance
1342:
1229:
1087:
992:
990:
658:
450:-space, an (open)
442:In Euclidean space
428:
395:
362:
342:
315:
288:
274:. In the field of
266:and informal use,
264:Euclidean geometry
182:(excluding them).
44:
3297:Equation 5.19.4,
3119:Spherical regions
2987:is any subset of
2970:topological space
2415:{\displaystyle r}
2150:are denoted with
2075:{\displaystyle y}
2055:{\displaystyle r}
2035:{\displaystyle r}
2015:{\displaystyle y}
1995:{\displaystyle x}
1755:
1711:{\displaystyle X}
1661:
1587:
1496:of the open ball
1480:
1090:{\displaystyle r}
963:
866:
748:
643:
629:
609:
365:{\displaystyle n}
291:{\displaystyle n}
241:Euclidean 3-space
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138:
120:
16:(Redirected from
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3420:
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3247:
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3214:, or hypersphere
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3024:and to the open
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2986:
2981:topological ball
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1511:
1491:
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1483:
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1455:
1443:
1388:taxicab distance
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1283:
1267:
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1022:
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1006:double factorial
1001:
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991:
983:
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958:
954:
935:
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933:
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907:
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867:
865:
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807:
789:
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761:
749:
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729:
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710:
671:
667:
665:
664:
659:
654:
653:
644:
642:
641:
637:
630:
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611:
610:
602:
596:
582:
581:
565:
559:
553:
535:
524:
507:
492:
485:
481:
477:
474:-ball of radius
473:
469:
465:
461:
457:
454:-ball of radius
453:
449:
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429:
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426:
404:
402:
401:
396:
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371:
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343:
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324:
322:
321:
316:
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297:
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224:
209:
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141:
134:
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127:
121:
119:
78:
54:
46:
21:
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3459:
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3454:
3453:
3439:Metric geometry
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3105:to a Euclidean
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3055:
3051:
3044:
3040:
3033:
3027:
3019:
3015:
3000:
2996:
2988:
2984:
2976:
2972:
2966:
2955:
2951:
2945:
2935:
2931:
2912:
2880:
2876:
2853:
2849:
2830:
2819:
2818:
2793:
2792:
2782:
2779:superellipsoids
2771:
2768:
2764:
2757:
2751:
2744:
2738:
2731:
2720:
2717:
2711:
2700:
2694:
2677:
2671:
2648:
2643:
2642:
2635:
2579:
2569:
2543:
2533:
2513:
2503:
2497:
2493:
2492:
2472:
2471:
2455:
2448:
2444:
2424:
2423:
2404:
2403:
2358:
2348:
2322:
2312:
2292:
2282:
2276:
2272:
2271:
2251:
2250:
2245:
2244:
2219:
2218:
2215:
2211:
2206:
2199:
2197:Cartesian space
2193:
2188:
2152:
2151:
2126:
2125:
2092:
2087:
2086:
2064:
2063:
2044:
2043:
2024:
2023:
2004:
2003:
2002:around a point
1984:
1983:
1953:
1948:
1947:
1892:
1891:
1866:
1865:
1861:
1855:
1823:
1822:
1786:
1781:
1780:
1732:
1731:
1724:
1723:
1720:discrete metric
1700:
1699:
1666:
1638:
1637:
1630:
1629:
1592:
1564:
1563:
1539:
1534:
1533:
1503:
1498:
1497:
1457:
1456:
1449:
1448:
1441:
1424:can serve as a
1415:totally bounded
1359:
1355:
1275:
1270:
1269:
1265:
1261:
1255:
1251:
1246:
1162:
1157:
1156:
1152:
1148:
1144:
1129:
1120:
1115:
1111:
1107:
1103:
1079:
1078:
1076:
1072:
1064:
1063:, namely a set
1048:
1045:
1024:
1016:
1008:
989:
988:
965:
941:
937:
936:
917:
913:
912:
900:
896:
892:
868:
841:
837:
836:
825:
809:
808:
799:
771:
768:
767:
750:
740:
730:
721:
699:
690:
689:
669:
645:
620:
616:
612:
597:
573:
568:
567:
561:
555:
551:
548:
542:
530:
519:
502:
490:
483:
479:
475:
471:
467:
463:
459:
455:
451:
447:
444:
418:
407:
406:
385:
374:
373:
354:
353:
352:while the open
332:
327:
326:
305:
300:
299:
280:
279:
229:Euclidean plane
219:
207:
198:
187:Euclidean space
176:boundary points
174:(including the
142:
131:
125:
122:
79:
77:
67:
55:
36:Euclidean space
28:
23:
22:
15:
12:
11:
5:
3462:
3460:
3452:
3451:
3446:
3441:
3436:
3426:
3425:
3422:
3421:
3403:(4): 277–283.
3390:
3361:hep-th/9506042
3354:(4): 585–610.
3341:
3323:(2): 101–107.
3307:
3306:
3290:
3283:
3258:
3257:
3255:
3252:
3250:
3249:
3244:
3234:
3225:
3220:
3215:
3206:
3201:
3195:
3190:
3184:
3179:
3172:
3170:
3167:
3166:
3165:
3157:
3149:
3141:
3133:
3114:
3111:
3093:A topological
3009:cell complexes
2965:
2962:
2911:
2908:
2896:
2892:
2887:
2883:
2879:
2875:
2872:
2869:
2865:
2860:
2856:
2852:
2848:
2845:
2842:
2837:
2833:
2829:
2826:
2806:
2803:
2800:
2766:
2755:
2748:body diagonals
2742:
2715:
2698:
2675:
2657:
2652:
2623:
2619:
2615:
2612:
2607:
2603:
2599:
2594:
2588:
2583:
2576:
2572:
2567:
2563:
2560:
2557:
2552:
2547:
2540:
2536:
2531:
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2522:
2517:
2510:
2506:
2501:
2496:
2491:
2486:
2481:
2478:
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2470:
2464:
2459:
2454:
2451:
2447:
2443:
2440:
2437:
2434:
2431:
2411:
2391:
2386:
2382:
2378:
2373:
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2362:
2355:
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2336:
2331:
2326:
2319:
2315:
2310:
2306:
2301:
2296:
2289:
2285:
2280:
2275:
2270:
2265:
2260:
2257:
2254:
2232:
2229:
2226:
2213:
2192:
2186:
2171:
2168:
2165:
2162:
2159:
2139:
2136:
2133:
2113:
2110:
2107:
2104:
2099:
2095:
2071:
2051:
2031:
2011:
1991:
1971:
1968:
1965:
1960:
1956:
1935:
1932:
1929:
1926:
1923:
1920:
1917:
1914:
1911:
1908:
1905:
1902:
1899:
1879:
1876:
1873:
1854:
1851:
1839:
1836:
1833:
1830:
1810:
1807:
1804:
1801:
1798:
1793:
1789:
1768:
1765:
1762:
1759:
1754:
1750:
1747:
1744:
1739:
1735:
1707:
1687:
1684:
1681:
1678:
1673:
1669:
1665:
1660:
1656:
1653:
1650:
1645:
1641:
1627:
1613:
1610:
1607:
1604:
1599:
1595:
1591:
1586:
1582:
1579:
1576:
1571:
1567:
1560:
1557:
1554:
1551:
1546:
1542:
1521:
1518:
1515:
1510:
1506:
1479:
1475:
1472:
1469:
1464:
1460:
1392:cross-polytope
1341:
1338:
1335:
1332:
1329:
1326:
1323:
1320:
1317:
1314:
1311:
1308:
1305:
1302:
1299:
1296:
1293:
1290:
1287:
1282:
1278:
1249:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1169:
1165:
1118:
1101:ball of radius
1086:
1044:
1041:
987:
981:
978:
975:
972:
968:
961:
957:
953:
950:
947:
944:
940:
932:
927:
923:
920:
916:
910:
906:
903:
899:
895:
889:
884:
881:
878:
875:
871:
864:
861:
857:
853:
850:
847:
844:
840:
832:
828:
822:
819:
816:
812:
805:
802:
800:
798:
795:
792:
787:
784:
781:
778:
774:
770:
769:
766:
760:
757:
753:
746:
743:
737:
733:
727:
724:
722:
720:
717:
714:
709:
706:
702:
698:
697:
678:gamma function
674:Leonhard Euler
657:
652:
648:
640:
636:
633:
628:
625:
619:
615:
608:
605:
600:
594:
591:
588:
585:
580:
576:
544:Main article:
541:
538:
443:
440:
425:
421:
417:
414:
392:
388:
384:
381:
361:
339:
335:
312:
308:
287:
255:, a ball is a
193:in general. A
170:. It may be a
144:
143:
58:
56:
49:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3461:
3450:
3447:
3445:
3442:
3440:
3437:
3435:
3432:
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3429:
3418:
3414:
3410:
3406:
3402:
3398:
3397:
3391:
3387:
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3375:
3371:
3367:
3362:
3357:
3353:
3349:
3348:
3342:
3338:
3334:
3330:
3326:
3322:
3318:
3317:
3311:
3310:
3303:
3300:
3294:
3291:
3286:
3284:9780262590204
3280:
3276:
3272:
3271:
3263:
3260:
3253:
3243:
3238:
3235:
3233:
3228:Volume of an
3226:
3224:
3221:
3219:
3216:
3213:
3207:
3205:
3202:
3199:
3196:
3194:
3191:
3188:
3185:
3183:
3180:
3177:
3174:
3173:
3168:
3163:
3162:
3158:
3155:
3154:
3150:
3147:
3146:
3142:
3139:
3138:
3134:
3131:
3130:
3126:
3125:
3124:
3120:
3112:
3110:
3104:
3103:diffeomorphic
3100:
3091:
3085:
3080:
3066:
3062:
3048:
3037:
3031:
3022:
3012:
3010:
3006:
2994:
2982:
2979:-dimensional
2971:
2963:
2961:
2958:
2948:
2943:
2938:
2929:
2925:
2921:
2917:
2909:
2907:
2890:
2885:
2881:
2877:
2873:
2870:
2867:
2863:
2858:
2854:
2850:
2840:
2827:
2801:
2798:
2789:
2785:
2780:
2774:
2762:
2754:
2749:
2741:
2734:
2728:
2726:
2714:
2709:
2705:
2697:
2692:
2688:
2684:
2683:
2674:
2655:
2638:
2621:
2617:
2613:
2610:
2605:
2601:
2597:
2592:
2586:
2574:
2570:
2561:
2558:
2555:
2550:
2538:
2534:
2525:
2520:
2508:
2504:
2494:
2489:
2484:
2476:
2468:
2462:
2452:
2449:
2445:
2441:
2435:
2429:
2409:
2389:
2384:
2380:
2376:
2371:
2365:
2353:
2349:
2340:
2337:
2334:
2329:
2317:
2313:
2304:
2299:
2287:
2283:
2273:
2268:
2263:
2255:
2230:
2227:
2224:
2210:
2202:
2198:
2187:
2185:
2182:
2169:
2163:
2157:
2137:
2134:
2131:
2111:
2105:
2097:
2093:
2085:
2069:
2049:
2029:
2009:
1989:
1966:
1958:
1954:
1933:
1927:
1924:
1921:
1915:
1909:
1906:
1903:
1897:
1874:
1860:
1852:
1850:
1837:
1834:
1831:
1828:
1808:
1805:
1799:
1791:
1787:
1763:
1757:
1745:
1737:
1733:
1721:
1705:
1685:
1679:
1671:
1667:
1663:
1651:
1643:
1639:
1625:
1611:
1605:
1597:
1593:
1589:
1577:
1569:
1565:
1558:
1552:
1544:
1540:
1516:
1508:
1504:
1495:
1470:
1462:
1458:
1445:
1439:
1435:
1431:
1427:
1423:
1418:
1416:
1412:
1407:
1405:
1401:
1397:
1393:
1389:
1385:
1381:
1377:
1372:
1370:
1369:
1362:
1352:
1339:
1333:
1330:
1324:
1321:
1318:
1312:
1309:
1306:
1303:
1300:
1294:
1288:
1280:
1276:
1258:
1252:
1244:
1239:
1226:
1220:
1217:
1211:
1208:
1205:
1199:
1196:
1193:
1190:
1187:
1181:
1175:
1167:
1163:
1140:
1136:
1132:
1125:
1121:
1102:
1084:
1070:
1062:
1056:
1052:
1042:
1040:
1036:
1032:
1028:
1020:
1012:
1007:
1002:
985:
979:
976:
973:
970:
966:
959:
955:
951:
948:
945:
942:
938:
930:
925:
921:
918:
914:
908:
904:
901:
897:
893:
887:
882:
879:
876:
873:
869:
862:
859:
855:
851:
848:
845:
842:
838:
830:
826:
820:
817:
814:
810:
803:
801:
793:
785:
782:
779:
776:
772:
764:
758:
755:
751:
744:
741:
735:
731:
725:
723:
715:
707:
704:
700:
687:
683:
679:
675:
655:
650:
646:
638:
634:
631:
626:
623:
617:
606:
603:
598:
592:
586:
578:
574:
564:
558:
547:
539:
537:
533:
528:
522:
517:
514:bounded by a
513:
512:
505:
500:
496:
489:In Euclidean
487:
446:In Euclidean
441:
439:
423:
419:
415:
412:
390:
386:
382:
379:
359:
337:
333:
310:
306:
285:
277:
273:
269:
265:
260:
258:
254:
250:
247:bounded by a
246:
242:
238:
234:
230:
226:
222:
215:
211:
204:
196:
192:
191:metric spaces
188:
183:
181:
177:
173:
169:
165:
164:
160:bounded by a
159:
155:
151:
140:
137:
129:
118:
115:
111:
108:
104:
101:
97:
94:
90:
87: –
86:
82:
81:Find sources:
75:
71:
65:
64:
59:This article
57:
53:
48:
47:
41:
37:
32:
19:
3400:
3394:
3351:
3345:
3320:
3314:
3298:
3293:
3269:
3262:
3241:
3159:
3151:
3143:
3135:
3127:
3122:
3092:
3078:
3064:
3060:
3049:
3035:
3032:(hypercube)
3020:
3013:
2993:homeomorphic
2980:
2967:
2956:
2946:
2936:
2913:
2790:
2783:
2772:
2760:
2752:
2747:
2739:
2732:
2729:
2712:
2707:
2695:
2690:
2686:
2680:
2672:
2636:
2200:
2194:
2183:
2083:
2082:) copy of a
1856:
1446:
1437:
1422:metric space
1419:
1408:
1373:
1366:
1360:
1353:
1256:
1247:
1242:
1240:
1138:
1134:
1130:
1123:
1116:
1100:
1061:metric space
1054:
1050:
1046:
1034:
1030:
1026:
1018:
1010:
1003:
562:
556:
549:
531:
520:
509:
503:
488:
445:
271:
267:
261:
257:line segment
220:
213:
206:
202:
194:
184:
179:
171:
168:solid sphere
167:
161:
158:solid figure
153:
147:
132:
123:
113:
106:
99:
92:
80:
68:Please help
63:verification
60:
39:
3187:Formal ball
2725:Lamé curves
2243:and defines
1492:denote the
1440:the metric
668:where
495:hypersphere
470:. A closed
458:and center
278:the closed
214:hypersphere
172:closed ball
150:mathematics
18:Closed ball
3428:Categories
3254:References
3237:Octahedron
3117:See also:
1982:of points
1864:with norm
1722:, one has
1378:under the
1264:away from
1151:away from
1075:, and let
482:away from
126:March 2024
96:newspapers
3417:119483499
3386:119438515
3275:MIT Press
3034:(0, 1) ⊆
2991:which is
2871:…
2836:∞
2832:‖
2825:‖
2805:∞
2704:Chebyshev
2691:diagonals
2687:Manhattan
2559:⋯
2453:∈
2338:⋯
2228:≥
2204:with the
2084:unit ball
1931:‖
1925:−
1919:‖
1878:‖
1875:⋅
1872:‖
1832:∈
1753:¯
1718:with the
1659:¯
1590:⊆
1585:¯
1559:⊆
1478:¯
1384:hypercube
1368:unit ball
1331:≤
1310:∣
1304:∈
1197:∣
1191:∈
1033:− 1) ⋅ (2
922:π
827:π
732:π
682:factorial
614:Γ
599:π
416:
383:
203:hyperball
180:open ball
3449:Topology
3223:Manifold
3204:3-sphere
3169:See also
3086:of
3047:-cube .
2480:‖
2474:‖
2259:‖
2253:‖
1821:for any
1430:topology
499:interval
276:topology
225:)-sphere
3444:Spheres
3366:Bibcode
3337:2690391
3248:metric.
3212:-sphere
3145:segment
3113:Regions
3109:-ball.
2930:subset
2920:bounded
2682:taxicab
1494:closure
1411:bounded
1396:compact
1097:
1077:
1067:with a
508:, is a
251:. In a
156:is the
110:scholar
3415:
3384:
3335:
3281:
3198:Sphere
3137:sector
3099:smooth
3072:-ball
2928:convex
2926:, and
2775:> 2
2750:, the
2737:, the
1624:it is
1434:unions
1404:convex
1363:> 0
1243:closed
1069:metric
1013:+ 1)!!
540:Volume
527:sphere
516:circle
268:sphere
245:volume
237:circle
163:sphere
112:
105:
98:
91:
83:
3434:Balls
3413:S2CID
3382:S2CID
3356:arXiv
3333:JSTOR
3232:-ball
3161:wedge
3153:shell
3030:-cube
3026:unit
2770:with
2761:edges
2708:sides
2209:-norm
2195:In a
2191:-norm
1390:is a
1382:is a
1059:be a
529:when
518:when
501:when
466:from
239:. In
210:-ball
117:JSTOR
103:books
3279:ISBN
3176:Ball
3076:and
2942:norm
2924:open
2777:are
2730:For
2634:For
2611:<
1857:Any
1779:but
1447:Let
1426:base
1365:. A
1241:The
1218:<
1047:Let
1037:+ 1)
550:The
511:disk
272:ball
233:disk
195:ball
154:ball
152:, a
89:news
40:ball
38:, a
3405:doi
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2010:y
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