978:, in which there is essentially only one extremiser up to affine transformations (namely the ball), there are many distinct extremisers for the lower bound - not only the cube and the octahedron, but also products of cubes and octahedra, polar bodies of products of cubes and octahedra, products of polar bodies of… well, you get the idea. It is really difficult to conceive of any sort of flow or optimisation procedure which would converge to exactly these bodies and no others; a radically different type of argument might be needed.
2644:
1148:
The Mahler volume can be defined in the same way, as the product of the volume and the polar volume, for convex bodies whose interior contains the origin regardless of symmetry. Mahler conjectured that, for this generalization, the minimum volume is obtained by a
697:
237:
1153:, with its centroid at the origin. As with the symmetric Mahler conjecture, reverse SantalĂł inequalities are known showing that the minimum volume is at least within an exponential factor of the simplex.
406:
867:
820:
1755:
542:
1075:
1525:
932:
which include these two types of shapes, as well as their affine transformations. The Mahler conjecture states that the Mahler volume of these shapes is the smallest of any
496:
304:
172:
960:
720:
1036:
469:
1010:
177:
1672:
893:
The
Blaschke–Santaló inequality states that the shapes with maximum Mahler volume are the spheres and ellipsoids. The three-dimensional case of this result was
595:
566:
446:
426:
327:
277:
257:
142:
122:
1748:
1112:
1124:= 0}. This was first proven by Saint-Raymond in 1980. Later, a much shorter proof was found by Meyer. This was further generalized to convex bodies with
2533:
17:
1741:
605:
1101:. This was first proven by Nazarov, Petrov, Ryabogin, and Zvavitch for the unit cube, and later generalized to all Hanner polytopes by Jaegil Kim.
1610:
Nazarov, Fedor; Petrov, Fedor; Ryabogin, Dmitry; Zvavitch, Artem (2010). "A remark on the Mahler conjecture: local minimality of the unit cube".
2369:
2196:
1132:. The minimizers are then not necessarily Hanner polytopes, but were found to be regular polytopes corresponding to the reflection groups.
2359:
1945:
2486:
2341:
887:
2317:
1845:
1701:
917:
by which any centrally symmetric convex body can be replaced with a more sphere-like body without decreasing its Mahler volume.
336:
1937:
1724:
1211:
Iriyeh, Hiroshi; Shibata, Masataka (2020). "Symmetric Mahler's conjecture for the volume product in the 3-dimensional case".
1090:
The 2-dimensional case of the Mahler conjecture has been solved by Mahler and the 3-dimensional case by Iriyeh and
Shibata.
828:
2209:
1950:
884:
Is the Mahler volume of a centrally symmetric convex body always at least that of the hypercube of the same dimension?
2298:
2189:
1571:
771:
2568:
1097:
is a strict local minimizer for the Mahler volume in the class of origin-symmetric convex bodies endowed with the
2673:
2213:
1970:
1887:
825:
The Mahler volume of the sphere is larger than the Mahler volume of the hypercube by a factor of approximately
2364:
1716:
1364:
2647:
2420:
2354:
2182:
1990:
501:
2384:
2629:
2583:
2507:
2389:
1955:
1139:
equal to zero almost everywhere on its boundary, suggesting strongly that a minimal body is a polytope.
1045:
1501:
2624:
2440:
1995:
1985:
1612:
1213:
1098:
1426:
Reisner, Shlomo; SchĂĽtt, Carsten; Werner, Elisabeth M. (2012). "Mahler's
Conjecture and Curvature".
1391:
Barthe, F.; Fradelizi, M. (April 2013). "The volume product of convex bodies with many symmetries".
2668:
2476:
2374:
2277:
1975:
1960:
1802:
1529:
2573:
2349:
2045:
2022:
1916:
1840:
1676:
50:
474:
282:
150:
2604:
2548:
2512:
2163:
2124:
2040:
1965:
1892:
1877:
1830:
975:
914:
545:
46:
2109:
2101:
2097:
2093:
2089:
2085:
2311:
1897:
939:
705:
54:
2307:
1015:
451:
2587:
1902:
1768:
1733:
1689:
1643:
1602:
1558:
1538:
1244:
988:
2174:
1482:(1917). "Uber affine Geometrie VII: Neue Extremeingenschaften von Ellipse und Ellipsoid".
599:
is itself another unit sphere. Thus, its Mahler volume is just the square of its volume,
8:
2553:
2491:
2205:
1835:
1825:
1820:
1125:
1542:
1498:; Milman, Vitali D. (1987). "New volume ratio properties for convex symmetric bodies in
2578:
2445:
2064:
1782:
1657:
1621:
1580:
1435:
1408:
1359:
1282:
1222:
1136:
894:
580:
551:
431:
411:
312:
262:
242:
127:
107:
62:
2558:
1882:
1720:
1038:, matching the scaling behavior of the hypercube volume but with a smaller constant.
2563:
2481:
2450:
2430:
2415:
2410:
2405:
2035:
1980:
1871:
1866:
1631:
1590:
1546:
1479:
1445:
1400:
1373:
1321:
1292:
1232:
1129:
1115:, that is, convex bodies invariant under reflection on each coordinate hyperplane {
898:
39:
2242:
2678:
2425:
2379:
2327:
2322:
2293:
2055:
2026:
2000:
1921:
1906:
1815:
1787:
1764:
1685:
1639:
1598:
1554:
1240:
1094:
929:
742:
90:
31:
2252:
1651:
906:
2614:
2466:
2267:
2142:
2050:
1911:
1810:
1566:
1236:
925:
762:
723:
692:{\displaystyle {\frac {\Gamma (3/2)^{2n}4^{n}}{\Gamma ({\frac {n}{2}}+1)^{2}}}}
1635:
1594:
1297:
1270:
61:. It is known that the shapes with the largest possible Mahler volume are the
2662:
2619:
2543:
2272:
2257:
2247:
1926:
1495:
1360:"Une caractérisation volumique de certains espaces normés de dimension finie"
1135:
Reisner et al. (2010) showed that a minimizer of the Mahler volume must have
746:
94:
232:{\displaystyle \left\{x\mid x\cdot y\leq 1{\text{ for all }}y\in B\right\}.}
2609:
2262:
2232:
2147:
1449:
1404:
1339:
Saint-Raymond, J. (1980). "Sur le volume des corps convexes symétriques".
174:
is another centrally symmetric body in the same space, defined as the set
2538:
2528:
2435:
2237:
2137:
2132:
2016:
1708:
1697:
1193:
963:
58:
42:
1412:
2471:
2303:
2060:
2030:
1792:
1550:
1377:
1325:
971:
876:
758:
734:
330:
97:
1585:
921:
754:
727:
145:
101:
78:
66:
1713:
Structure and
Randomness: Pages from Year One of a Mathematical Blog
2069:
1227:
1105:
738:
577:
1626:
1440:
1287:
1850:
1569:(2008). "From the Mahler conjecture to Gauss linking integrals".
1150:
77:
states that the minimum possible Mahler volume is attained by a
1312:
Reisner, Shlomo (1986). "Zonoids with minimal volume-product".
1609:
1257:
936:-dimensional symmetric convex body; it remains unsolved when
1763:
2204:
750:
1012:
times the volume of a sphere for some absolute constant
401:{\displaystyle (TB)^{\circ }=(T^{-1})^{\ast }B^{\circ }}
1702:"Open question: the Mahler conjecture on convex bodies"
1056:
838:
1711:(2009). "3.8 Mahler's conjecture for convex bodies".
1660:
1504:
1484:
Ber. Verh. Sächs. Akad. Wiss. Leipzig Math.-Phys. Kl.
1456:
1048:
1018:
991:
942:
831:
774:
708:
608:
583:
554:
504:
477:
454:
434:
414:
339:
315:
285:
265:
245:
180:
153:
130:
110:
1077:
in this bound. A result of this type is known as a
920:
The shapes with the minimum known Mahler volume are
1425:
1196:(1939). "Ein Minimalproblem fĂĽr konvexe Polygone".
862:{\displaystyle \left({\tfrac {\pi }{2}}\right)^{n}}
2534:Spectral theory of ordinary differential equations
1666:
1654:(1949). "An affine invariant for convex bodies of
1519:
1069:
1030:
1004:
985:proved that the Mahler volume is bounded below by
954:
861:
814:
714:
691:
589:
560:
536:
490:
463:
440:
420:
400:
321:
298:
271:
251:
231:
166:
136:
116:
57:. It is named after German-English mathematician
2660:
505:
455:
1390:
878:
1494:
1271:"Minimal volume product near Hanner polytopes"
982:
815:{\displaystyle {\frac {4^{n}}{\Gamma (n+1)}}.}
2190:
1749:
1338:
1210:
1111:The Mahler conjecture holds in the class of
905:); the full result was proven much later by
1042:proved that, more concretely, one can take
2197:
2183:
1756:
1742:
1428:International Mathematics Research Notices
1625:
1584:
1565:
1507:
1462:
1439:
1296:
1286:
1226:
1039:
974:is so difficult is that unlike the upper
765:. Its Mahler volume can be calculated as
2487:Group algebra of a locally compact group
1478:
1143:
902:
568:is preserved by linear transformations.
124:is a centrally symmetric convex body in
49:that is associated with the body and is
1846:Locally convex topological vector space
1650:
1311:
910:
888:(more unsolved problems in mathematics)
14:
2661:
1192:
2178:
1737:
1357:
1176:
1174:
1172:
1170:
1168:
1166:
749:. In particular, the polar body of a
537:{\displaystyle \det(T^{-1})^{\ast }}
1707:
1696:
1268:
1180:
24:
1341:SĂ©minaire d'initiation Ă l'analyse
1163:
1084:
788:
709:
654:
612:
144:-dimensional Euclidean space, the
25:
2690:
1572:Geometric and Functional Analysis
1070:{\displaystyle c={\tfrac {1}{2}}}
872:
259:is the product of the volumes of
2643:
2642:
2569:Topological quantum field theory
1520:{\displaystyle \mathbb {R} ^{n}}
1104:The Mahler conjecture holds for
1951:Ekeland's variational principle
1419:
1393:American Journal of Mathematics
879:Unsolved problem in mathematics
548:, the overall Mahler volume of
1384:
1351:
1332:
1305:
1275:Journal of Functional Analysis
1262:
1251:
1204:
1186:
803:
791:
677:
657:
630:
615:
525:
508:
379:
362:
350:
340:
13:
1:
2365:Uniform boundedness principle
1717:American Mathematical Society
1472:
1365:Israel Journal of Mathematics
1093:It is known that each of the
913:) using a technique known as
544:. As these determinants are
471:and multiplies the volume of
84:
983:Bourgain & Milman (1987)
730:has the same Mahler volume.
726:. By affine invariance, any
333:linear transformation, then
7:
1971:Hermite–Hadamard inequality
571:
71:Blaschke–Santaló inequality
69:; this is now known as the
18:Blaschke–Santaló inequality
10:
2695:
2508:Invariant subspace problem
1237:10.1215/00127094-2019-0072
1079:reverse SantalĂł inequality
491:{\displaystyle B^{\circ }}
299:{\displaystyle B^{\circ }}
167:{\displaystyle B^{\circ }}
27:Concept in convex geometry
2638:
2597:
2521:
2500:
2459:
2398:
2340:
2286:
2228:
2221:
2156:
2123:
2078:
2009:
1935:
1859:
1801:
1775:
1706:Revised and reprinted in
1636:10.1215/00127094-2010-042
1613:Duke Mathematical Journal
1595:10.1007/s00039-008-0669-4
1314:Mathematische Zeitschrift
1298:10.1016/j.jfa.2013.08.008
1214:Duke Mathematical Journal
970:The main reason why this
928:, and more generally the
448:multiplies its volume by
2477:Spectrum of a C*-algebra
2157:Applications and related
1961:Fenchel-Young inequality
1530:Inventiones Mathematicae
1156:
597:-dimensional unit sphere
2574:Noncommutative geometry
1917:Legendre transformation
1841:Legendre transformation
1677:Portugaliae Mathematica
1358:Meyer, Mathieu (1986).
1198:Mathematica (Zutphen) B
955:{\displaystyle n\geq 4}
715:{\displaystyle \Gamma }
546:multiplicative inverses
2674:Geometric inequalities
2630:Tomita–Takesaki theory
2605:Approximation property
2549:Calculus of variations
2164:Convexity in economics
2098:(lower) ideally convex
1956:Fenchel–Moreau theorem
1946:Carathéodory's theorem
1668:
1521:
1128:that are more general
1071:
1032:
1031:{\displaystyle c>0}
1006:
980:
956:
915:Steiner symmetrization
863:
816:
716:
693:
591:
562:
538:
492:
465:
464:{\displaystyle \det T}
442:
422:
402:
323:
300:
273:
253:
233:
168:
138:
118:
55:linear transformations
47:dimensionless quantity
2625:Banach–Mazur distance
2588:Generalized functions
2086:Convex series related
1986:Shapley–Folkman lemma
1674:-dimensional space".
1669:
1522:
1405:10.1353/ajm.2013.0018
1258:Nazarov et al. (2010)
1144:For asymmetric bodies
1099:Banach–Mazur distance
1072:
1033:
1007:
1005:{\displaystyle c^{n}}
968:
957:
864:
817:
717:
694:
592:
576:The polar body of an
563:
539:
493:
466:
443:
423:
403:
324:
301:
274:
254:
239:The Mahler volume of
234:
169:
139:
119:
73:. The still-unsolved
2370:Kakutani fixed-point
2355:Riesz representation
1976:Krein–Milman theorem
1769:variational analysis
1719:. pp. 216–219.
1658:
1502:
1269:Kim, Jaegil (2014).
1113:unconditional bodies
1046:
1016:
989:
940:
899:Wilhelm Blaschke
829:
772:
733:The polar body of a
706:
606:
581:
552:
502:
475:
452:
432:
412:
337:
313:
283:
263:
243:
178:
151:
128:
108:
2554:Functional calculus
2513:Mahler's conjecture
2492:Von Neumann algebra
2206:Functional analysis
1966:Jensen's inequality
1836:Lagrange multiplier
1826:Convex optimization
1821:Convex metric space
1543:1987InMat..88..319B
1490:. Leipzig: 412–420.
1450:10.1093/imrn/rnr003
209: for all
40:centrally symmetric
2579:Riemann hypothesis
2278:Topological vector
2094:(cs, bcs)-complete
2065:Algebraic interior
1783:Convex combination
1664:
1551:10.1007/BF01388911
1517:
1378:10.1007/BF02765029
1326:10.1007/BF01164009
1137:Gaussian curvature
1067:
1065:
1028:
1002:
952:
859:
847:
812:
712:
689:
587:
558:
534:
488:
461:
438:
418:
398:
319:
296:
269:
249:
229:
164:
134:
114:
2656:
2655:
2559:Integral operator
2336:
2335:
2172:
2171:
1726:978-0-8218-4695-7
1700:(March 8, 2007).
1667:{\displaystyle n}
1480:Blaschke, Wilhelm
1130:reflection groups
1064:
846:
807:
687:
668:
590:{\displaystyle n}
561:{\displaystyle B}
441:{\displaystyle B}
421:{\displaystyle T}
322:{\displaystyle T}
272:{\displaystyle B}
252:{\displaystyle B}
210:
137:{\displaystyle n}
117:{\displaystyle B}
89:A convex body in
75:Mahler conjecture
16:(Redirected from
2686:
2646:
2645:
2564:Jones polynomial
2482:Operator algebra
2226:
2225:
2199:
2192:
2185:
2176:
2175:
2090:(cs, lcs)-closed
2036:Effective domain
1991:Robinson–Ursescu
1867:Convex conjugate
1758:
1751:
1744:
1735:
1734:
1730:
1705:
1693:
1673:
1671:
1670:
1665:
1652:SantalĂł, Luis A.
1647:
1629:
1606:
1588:
1562:
1526:
1524:
1523:
1518:
1516:
1515:
1510:
1491:
1466:
1463:Kuperberg (2008)
1460:
1454:
1453:
1443:
1423:
1417:
1416:
1388:
1382:
1381:
1355:
1349:
1348:
1336:
1330:
1329:
1309:
1303:
1302:
1300:
1290:
1281:(4): 2360–2402.
1266:
1260:
1255:
1249:
1248:
1230:
1221:(6): 1077–1134.
1208:
1202:
1201:
1190:
1184:
1178:
1095:Hanner polytopes
1076:
1074:
1073:
1068:
1066:
1057:
1040:Kuperberg (2008)
1037:
1035:
1034:
1029:
1011:
1009:
1008:
1003:
1001:
1000:
961:
959:
958:
953:
930:Hanner polytopes
907:Luis SantalĂł
880:
868:
866:
865:
860:
858:
857:
852:
848:
839:
821:
819:
818:
813:
808:
806:
786:
785:
776:
721:
719:
718:
713:
698:
696:
695:
690:
688:
686:
685:
684:
669:
661:
652:
651:
650:
641:
640:
625:
610:
596:
594:
593:
588:
567:
565:
564:
559:
543:
541:
540:
535:
533:
532:
523:
522:
497:
495:
494:
489:
487:
486:
470:
468:
467:
462:
447:
445:
444:
439:
427:
425:
424:
419:
407:
405:
404:
399:
397:
396:
387:
386:
377:
376:
358:
357:
328:
326:
325:
320:
305:
303:
302:
297:
295:
294:
278:
276:
275:
270:
258:
256:
255:
250:
238:
236:
235:
230:
225:
221:
211:
208:
173:
171:
170:
165:
163:
162:
143:
141:
140:
135:
123:
121:
120:
115:
93:is defined as a
21:
2694:
2693:
2689:
2688:
2687:
2685:
2684:
2683:
2669:Convex geometry
2659:
2658:
2657:
2652:
2634:
2598:Advanced topics
2593:
2517:
2496:
2455:
2421:Hilbert–Schmidt
2394:
2385:Gelfand–Naimark
2332:
2282:
2217:
2203:
2173:
2168:
2152:
2119:
2074:
2005:
1931:
1922:Semi-continuity
1907:Convex function
1888:Logarithmically
1855:
1816:Convex geometry
1797:
1788:Convex function
1771:
1765:Convex analysis
1762:
1727:
1659:
1656:
1655:
1567:Kuperberg, Greg
1511:
1506:
1505:
1503:
1500:
1499:
1475:
1470:
1469:
1461:
1457:
1424:
1420:
1389:
1385:
1356:
1352:
1337:
1333:
1310:
1306:
1267:
1263:
1256:
1252:
1209:
1205:
1191:
1187:
1179:
1164:
1159:
1146:
1126:symmetry groups
1123:
1087:
1085:Partial results
1055:
1047:
1044:
1043:
1017:
1014:
1013:
996:
992:
990:
987:
986:
941:
938:
937:
926:cross polytopes
891:
890:
885:
882:
875:
853:
837:
833:
832:
830:
827:
826:
787:
781:
777:
775:
773:
770:
769:
743:dual polyhedron
707:
704:
703:
680:
676:
660:
653:
646:
642:
633:
629:
621:
611:
609:
607:
604:
603:
582:
579:
578:
574:
553:
550:
549:
528:
524:
515:
511:
503:
500:
499:
482:
478:
476:
473:
472:
453:
450:
449:
433:
430:
429:
413:
410:
409:
392:
388:
382:
378:
369:
365:
353:
349:
338:
335:
334:
314:
311:
310:
290:
286:
284:
281:
280:
264:
261:
260:
244:
241:
240:
207:
185:
181:
179:
176:
175:
158:
154:
152:
149:
148:
129:
126:
125:
109:
106:
105:
91:Euclidean space
87:
32:convex geometry
28:
23:
22:
15:
12:
11:
5:
2692:
2682:
2681:
2676:
2671:
2654:
2653:
2651:
2650:
2639:
2636:
2635:
2633:
2632:
2627:
2622:
2617:
2615:Choquet theory
2612:
2607:
2601:
2599:
2595:
2594:
2592:
2591:
2581:
2576:
2571:
2566:
2561:
2556:
2551:
2546:
2541:
2536:
2531:
2525:
2523:
2519:
2518:
2516:
2515:
2510:
2504:
2502:
2498:
2497:
2495:
2494:
2489:
2484:
2479:
2474:
2469:
2467:Banach algebra
2463:
2461:
2457:
2456:
2454:
2453:
2448:
2443:
2438:
2433:
2428:
2423:
2418:
2413:
2408:
2402:
2400:
2396:
2395:
2393:
2392:
2390:Banach–Alaoglu
2387:
2382:
2377:
2372:
2367:
2362:
2357:
2352:
2346:
2344:
2338:
2337:
2334:
2333:
2331:
2330:
2325:
2320:
2318:Locally convex
2315:
2301:
2296:
2290:
2288:
2284:
2283:
2281:
2280:
2275:
2270:
2265:
2260:
2255:
2250:
2245:
2240:
2235:
2229:
2223:
2219:
2218:
2202:
2201:
2194:
2187:
2179:
2170:
2169:
2167:
2166:
2160:
2158:
2154:
2153:
2151:
2150:
2145:
2143:Strong duality
2140:
2135:
2129:
2127:
2121:
2120:
2118:
2117:
2082:
2080:
2076:
2075:
2073:
2072:
2067:
2058:
2053:
2051:John ellipsoid
2048:
2043:
2038:
2033:
2019:
2013:
2011:
2007:
2006:
2004:
2003:
1998:
1993:
1988:
1983:
1978:
1973:
1968:
1963:
1958:
1953:
1948:
1942:
1940:
1938:results (list)
1933:
1932:
1930:
1929:
1924:
1919:
1914:
1912:Invex function
1909:
1900:
1895:
1890:
1885:
1880:
1874:
1869:
1863:
1861:
1857:
1856:
1854:
1853:
1848:
1843:
1838:
1833:
1828:
1823:
1818:
1813:
1811:Choquet theory
1807:
1805:
1799:
1798:
1796:
1795:
1790:
1785:
1779:
1777:
1776:Basic concepts
1773:
1772:
1761:
1760:
1753:
1746:
1738:
1732:
1731:
1725:
1694:
1680:(in Spanish).
1663:
1648:
1620:(3): 419–430.
1607:
1579:(3): 870–892.
1563:
1537:(2): 319–340.
1514:
1509:
1496:Bourgain, Jean
1492:
1474:
1471:
1468:
1467:
1455:
1418:
1399:(2): 311–347.
1383:
1372:(3): 317–326.
1350:
1331:
1320:(3): 339–346.
1304:
1261:
1250:
1203:
1185:
1161:
1160:
1158:
1155:
1145:
1142:
1141:
1140:
1133:
1119:
1109:
1102:
1091:
1086:
1083:
1063:
1060:
1054:
1051:
1027:
1024:
1021:
999:
995:
951:
948:
945:
886:
883:
877:
874:
873:Extreme shapes
871:
856:
851:
845:
842:
836:
823:
822:
811:
805:
802:
799:
796:
793:
790:
784:
780:
763:cross polytope
724:Gamma function
711:
700:
699:
683:
679:
675:
672:
667:
664:
659:
656:
649:
645:
639:
636:
632:
628:
624:
620:
617:
614:
586:
573:
570:
557:
531:
527:
521:
518:
514:
510:
507:
485:
481:
460:
457:
437:
417:
395:
391:
385:
381:
375:
372:
368:
364:
361:
356:
352:
348:
345:
342:
318:
293:
289:
268:
248:
228:
224:
220:
217:
214:
206:
203:
200:
197:
194:
191:
188:
184:
161:
157:
133:
113:
104:interior. If
86:
83:
26:
9:
6:
4:
3:
2:
2691:
2680:
2677:
2675:
2672:
2670:
2667:
2666:
2664:
2649:
2641:
2640:
2637:
2631:
2628:
2626:
2623:
2621:
2620:Weak topology
2618:
2616:
2613:
2611:
2608:
2606:
2603:
2602:
2600:
2596:
2589:
2585:
2582:
2580:
2577:
2575:
2572:
2570:
2567:
2565:
2562:
2560:
2557:
2555:
2552:
2550:
2547:
2545:
2544:Index theorem
2542:
2540:
2537:
2535:
2532:
2530:
2527:
2526:
2524:
2520:
2514:
2511:
2509:
2506:
2505:
2503:
2501:Open problems
2499:
2493:
2490:
2488:
2485:
2483:
2480:
2478:
2475:
2473:
2470:
2468:
2465:
2464:
2462:
2458:
2452:
2449:
2447:
2444:
2442:
2439:
2437:
2434:
2432:
2429:
2427:
2424:
2422:
2419:
2417:
2414:
2412:
2409:
2407:
2404:
2403:
2401:
2397:
2391:
2388:
2386:
2383:
2381:
2378:
2376:
2373:
2371:
2368:
2366:
2363:
2361:
2358:
2356:
2353:
2351:
2348:
2347:
2345:
2343:
2339:
2329:
2326:
2324:
2321:
2319:
2316:
2313:
2309:
2305:
2302:
2300:
2297:
2295:
2292:
2291:
2289:
2285:
2279:
2276:
2274:
2271:
2269:
2266:
2264:
2261:
2259:
2256:
2254:
2251:
2249:
2246:
2244:
2241:
2239:
2236:
2234:
2231:
2230:
2227:
2224:
2220:
2215:
2211:
2207:
2200:
2195:
2193:
2188:
2186:
2181:
2180:
2177:
2165:
2162:
2161:
2159:
2155:
2149:
2146:
2144:
2141:
2139:
2136:
2134:
2131:
2130:
2128:
2126:
2122:
2115:
2113:
2107:
2105:
2099:
2095:
2091:
2087:
2084:
2083:
2081:
2077:
2071:
2068:
2066:
2062:
2059:
2057:
2054:
2052:
2049:
2047:
2044:
2042:
2039:
2037:
2034:
2032:
2028:
2024:
2020:
2018:
2015:
2014:
2012:
2008:
2002:
1999:
1997:
1994:
1992:
1989:
1987:
1984:
1982:
1981:Mazur's lemma
1979:
1977:
1974:
1972:
1969:
1967:
1964:
1962:
1959:
1957:
1954:
1952:
1949:
1947:
1944:
1943:
1941:
1939:
1934:
1928:
1927:Subderivative
1925:
1923:
1920:
1918:
1915:
1913:
1910:
1908:
1904:
1901:
1899:
1896:
1894:
1891:
1889:
1886:
1884:
1881:
1879:
1875:
1873:
1870:
1868:
1865:
1864:
1862:
1858:
1852:
1849:
1847:
1844:
1842:
1839:
1837:
1834:
1832:
1829:
1827:
1824:
1822:
1819:
1817:
1814:
1812:
1809:
1808:
1806:
1804:
1803:Topics (list)
1800:
1794:
1791:
1789:
1786:
1784:
1781:
1780:
1778:
1774:
1770:
1766:
1759:
1754:
1752:
1747:
1745:
1740:
1739:
1736:
1728:
1722:
1718:
1714:
1710:
1703:
1699:
1695:
1691:
1687:
1683:
1679:
1678:
1661:
1653:
1649:
1645:
1641:
1637:
1633:
1628:
1623:
1619:
1615:
1614:
1608:
1604:
1600:
1596:
1592:
1587:
1582:
1578:
1574:
1573:
1568:
1564:
1560:
1556:
1552:
1548:
1544:
1540:
1536:
1532:
1531:
1512:
1497:
1493:
1489:
1486:(in German).
1485:
1481:
1477:
1476:
1464:
1459:
1451:
1447:
1442:
1437:
1433:
1429:
1422:
1414:
1410:
1406:
1402:
1398:
1394:
1387:
1379:
1375:
1371:
1367:
1366:
1361:
1354:
1346:
1342:
1335:
1327:
1323:
1319:
1315:
1308:
1299:
1294:
1289:
1284:
1280:
1276:
1272:
1265:
1259:
1254:
1246:
1242:
1238:
1234:
1229:
1224:
1220:
1216:
1215:
1207:
1199:
1195:
1189:
1182:
1177:
1175:
1173:
1171:
1169:
1167:
1162:
1154:
1152:
1138:
1134:
1131:
1127:
1122:
1118:
1114:
1110:
1107:
1103:
1100:
1096:
1092:
1089:
1088:
1082:
1080:
1061:
1058:
1052:
1049:
1041:
1025:
1022:
1019:
997:
993:
984:
979:
977:
973:
967:
965:
949:
946:
943:
935:
931:
927:
923:
918:
916:
912:
908:
904:
900:
896:
889:
870:
854:
849:
843:
840:
834:
809:
800:
797:
794:
782:
778:
768:
767:
766:
764:
760:
756:
752:
748:
747:dual polytope
744:
740:
736:
731:
729:
725:
681:
673:
670:
665:
662:
647:
643:
637:
634:
626:
622:
618:
602:
601:
600:
598:
584:
569:
555:
547:
529:
519:
516:
512:
483:
479:
458:
435:
415:
393:
389:
383:
373:
370:
366:
359:
354:
346:
343:
332:
316:
307:
291:
287:
266:
246:
226:
222:
218:
215:
212:
204:
201:
198:
195:
192:
189:
186:
182:
159:
155:
147:
131:
111:
103:
99:
96:
92:
82:
80:
76:
72:
68:
64:
60:
56:
52:
48:
44:
41:
37:
36:Mahler volume
33:
19:
2610:Balanced set
2584:Distribution
2522:Applications
2375:Krein–Milman
2360:Closed graph
2148:Weak duality
2111:
2103:
2023:Orthogonally
1712:
1709:Tao, Terence
1698:Tao, Terence
1681:
1675:
1617:
1611:
1586:math/0610904
1576:
1570:
1534:
1528:
1487:
1483:
1458:
1431:
1427:
1421:
1396:
1392:
1386:
1369:
1363:
1353:
1344:
1340:
1334:
1317:
1313:
1307:
1278:
1274:
1264:
1253:
1218:
1212:
1206:
1197:
1194:Mahler, Kurt
1188:
1147:
1120:
1116:
1078:
981:
969:
933:
919:
892:
824:
732:
701:
575:
308:
88:
74:
70:
35:
29:
2539:Heat kernel
2529:Hardy space
2436:Trace class
2350:Hahn–Banach
2312:Topological
2138:Duality gap
2133:Dual system
2017:Convex hull
1684:: 155–161.
1434:(1): 1–16.
408:. Applying
59:Kurt Mahler
43:convex body
2663:Categories
2472:C*-algebra
2287:Properties
2061:Radial set
2031:Convex set
1793:Convex set
1473:References
1228:1706.01749
1200:: 118–127.
1181:Tao (2007)
972:conjecture
922:hypercubes
759:octahedron
735:polyhedron
331:invertible
146:polar body
98:convex set
85:Definition
67:ellipsoids
65:and solid
2446:Unbounded
2441:Transpose
2399:Operators
2328:Separable
2323:Reflexive
2308:Algebraic
2294:Barrelled
2046:Hypograph
1627:0905.0867
1441:1009.3583
1288:1212.2544
1106:zonotopes
964:Terry Tao
947:≥
841:π
789:Γ
755:hypercube
728:ellipsoid
710:Γ
655:Γ
613:Γ
530:∗
517:−
484:∘
394:∘
384:∗
371:−
355:∘
292:∘
216:∈
202:≤
196:⋅
190:∣
160:∘
102:non-empty
79:hypercube
51:invariant
2648:Category
2460:Algebras
2342:Theorems
2299:Complete
2268:Schwartz
2214:glossary
2070:Zonotope
2041:Epigraph
1413:23525797
966:writes:
739:polytope
572:Examples
2451:Unitary
2431:Nuclear
2416:Compact
2411:Bounded
2406:Adjoint
2380:Min–max
2273:Sobolev
2258:Nuclear
2248:Hilbert
2243:Fréchet
2208: (
2125:Duality
2027:Pseudo-
2001:Ursescu
1898:Pseudo-
1872:Concave
1851:Simplex
1831:Duality
1690:0039293
1644:2730574
1603:2438998
1559:0880954
1539:Bibcode
1245:4085078
1151:simplex
909: (
901: (
741:is its
722:is the
95:compact
2679:Volume
2426:Normal
2263:Orlicz
2253:Hölder
2233:Banach
2222:Spaces
2210:topics
2108:, and
2079:Series
1996:Simons
1903:Quasi-
1893:Proper
1878:Closed
1723:
1688:
1642:
1601:
1557:
1411:
1243:
895:proven
757:is an
702:where
329:is an
53:under
34:, the
2238:Besov
1936:Main
1622:arXiv
1581:arXiv
1436:arXiv
1409:JSTOR
1283:arXiv
1223:arXiv
1157:Notes
976:bound
962:. As
100:with
63:balls
45:is a
38:of a
2586:(or
2304:Dual
2056:Lens
2010:Sets
1860:Maps
1767:and
1721:ISBN
1432:2012
1023:>
911:1949
903:1917
751:cube
279:and
2110:(Hw
1632:doi
1618:154
1591:doi
1547:doi
1527:".
1446:doi
1401:doi
1397:135
1374:doi
1322:doi
1318:192
1293:doi
1279:266
1233:doi
1219:169
897:by
761:or
753:or
745:or
737:or
506:det
498:by
456:det
428:to
309:If
30:In
2665::
2212:–
2102:(H
2100:,
2096:,
2092:,
2029:)
2025:,
1905:)
1883:K-
1715:.
1686:MR
1640:MR
1638:.
1630:.
1616:.
1599:MR
1597:.
1589:.
1577:18
1575:.
1555:MR
1553:.
1545:.
1535:88
1533:.
1488:69
1444:.
1430:.
1407:.
1395:.
1370:55
1368:.
1362:.
1345:81
1343:.
1316:.
1291:.
1277:.
1273:.
1241:MR
1239:.
1231:.
1217:.
1165:^
1081:.
924:,
869:.
306:.
81:.
2590:)
2314:)
2310:/
2306:(
2216:)
2198:e
2191:t
2184:v
2116:)
2114:)
2112:x
2106:)
2104:x
2088:(
2063:/
2021:(
1876:(
1757:e
1750:t
1743:v
1729:.
1704:.
1692:.
1682:8
1662:n
1646:.
1634::
1624::
1605:.
1593::
1583::
1561:.
1549::
1541::
1513:n
1508:R
1465:.
1452:.
1448::
1438::
1415:.
1403::
1380:.
1376::
1347:.
1328:.
1324::
1301:.
1295::
1285::
1247:.
1235::
1225::
1183:.
1121:i
1117:x
1108:.
1062:2
1059:1
1053:=
1050:c
1026:0
1020:c
998:n
994:c
950:4
944:n
934:n
881::
855:n
850:)
844:2
835:(
810:.
804:)
801:1
798:+
795:n
792:(
783:n
779:4
682:2
678:)
674:1
671:+
666:2
663:n
658:(
648:n
644:4
638:n
635:2
631:)
627:2
623:/
619:3
616:(
585:n
556:B
526:)
520:1
513:T
509:(
480:B
459:T
436:B
416:T
390:B
380:)
374:1
367:T
363:(
360:=
351:)
347:B
344:T
341:(
317:T
288:B
267:B
247:B
227:.
223:}
219:B
213:y
205:1
199:y
193:x
187:x
183:{
156:B
132:n
112:B
20:)
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