681:
426:
236:
114:
527:
272:
711:
487:
456:
141:
44:
731:
519:
264:
206:
67:
1058:
Proceedings of the
Conference on Convexity and Combinatorial Geometry (Univ. Oklahoma, Norman, Okla., 1971). Dept. Math., Univ. Oklahoma, Norman, Oklahoma
888:
907:
905:
Koldobsky, Alexander (1998a), "Intersection bodies, positive definite distributions, and the
Busemann-Petty problem",
849:
1138:
998:
955:
797:
showed that a centrally symmetric star-shaped body is an intersection body if and only if the function 1/||
916:
854:
182:
921:
676:{\displaystyle V_{n}(K)^{n-1}V_{n}(\Pi ^{\circ }K)\geq V_{n}(T^{n})^{n-1}V_{n}(\Pi ^{\circ }T^{n}),}
421:{\displaystyle V_{n}(K)^{n-1}V_{n}(\Pi ^{\circ }K)\leq V_{n}(B^{n})^{n-1}V_{n}(\Pi ^{\circ }B^{n}),}
211:
1101:
80:
242:
of its projection body. There are two remarkable affine isoperimetric inequality for this body.
1017:
974:
934:
884:
166:
1110:
1085:
1007:
964:
926:
876:
117:
1122:
1065:
1049:
1029:
986:
946:
898:
689:
465:
434:
123:
26:
1118:
1061:
1045:
1025:
982:
942:
894:
872:
74:
17:
776:
716:
504:
249:
191:
174:
52:
805:|| is the homogeneous function of degree 1 that is 1 on the boundary of the body, and
1132:
1012:
494:
993:
784:
1037:
47:
1099:
Zhang, Gaoyong (1991), "Restricted chord projection and affine inequalities",
156:
1021:
978:
938:
239:
969:
930:
733:-dimensional simplex, and there is equality precisely for such simplices.
1114:
1090:
1073:
880:
829:
1074:"Zur einem Problem von Shephard ĂŒber die Projektionen konvexer Körper"
493:-dimensional volume, and there is equality precisely for ellipsoids.
747:
is defined similarly, as the star body such that for any vector
867:
Bourgain, Jean; Lindenstrauss, J. (1988), "Projection bodies",
1044:, Kobenhavns Univ. Mat. Inst., Copenhagen, pp. 234â241,
1042:
Proceedings of the
Colloquium on Convexity (Copenhagen, 1965)
771:. Equivalently, the radial function of the intersection body
871:, Lecture Notes in Math., vol. 1317, Berlin, New York:
169:
showed that the projection body of a convex body is convex.
953:
Koldobsky, Alexander (1998b), "Intersection bodies in RâŽ",
763: â 1)-dimensional volume of the intersection of
151: â 1)-dimensional volume of the projection of
996:(1988), "Intersection bodies and dual mixed volumes",
719:
692:
530:
507:
468:
437:
275:
252:
214:
194:
126:
83:
55:
29:
1056:
Petty, Clinton M. (1971), "Isoperimetric problems",
869:Geometric aspects of functional analysis (1986/87)
725:
705:
675:
513:
481:
450:
420:
258:
230:
200:
135:
108:
61:
38:
801:|| is a positive definite distribution, where ||
181:) used projection bodies in their solution to
8:
836:=4 but are not intersection bodies for
77:is the convex body such that for any vector
806:
794:
1089:
1011:
968:
920:
783:. Intersection bodies were introduced by
718:
697:
691:
661:
651:
638:
622:
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296:
280:
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178:
125:
94:
82:
54:
28:
809:used this to show that the unit balls l
788:
498:
243:
170:
7:
501:) proved that for all convex bodies
648:
577:
393:
322:
246:proved that for all convex bodies
216:
127:
30:
14:
908:American Journal of Mathematics
667:
644:
619:
605:
589:
573:
548:
541:
412:
389:
364:
350:
334:
318:
293:
286:
231:{\displaystyle \Pi ^{\circ }K}
1:
755:from the origin in direction
1013:10.1016/0001-8708(88)90077-1
832:are intersection bodies for
828:-dimensional space with the
109:{\displaystyle u\in S^{n-1}}
462:-dimensional unit ball and
1155:
1036:Petty, Clinton M. (1967),
779:of the radial function of
1078:Mathematische Zeitschrift
1072:Schneider, Rolf (1967).
999:Advances in Mathematics
956:Advances in Mathematics
751:the radial function of
970:10.1006/aima.1998.1718
850:BusemannâPetty problem
727:
707:
677:
515:
483:
452:
422:
260:
232:
202:
137:
110:
63:
40:
931:10.1353/ajm.1998.0030
728:
708:
706:{\displaystyle T^{n}}
678:
516:
484:
482:{\displaystyle V_{n}}
453:
451:{\displaystyle B^{n}}
423:
261:
233:
203:
138:
136:{\displaystyle \Pi K}
111:
64:
41:
39:{\displaystyle \Pi K}
875:, pp. 250â270,
767:with the hyperplane
717:
690:
528:
505:
466:
435:
273:
250:
212:
192:
124:
81:
53:
27:
1102:Geometriae Dedicata
1038:"Projection bodies"
820:, 2 <
208:a convex body, let
1115:10.1007/BF00182294
1091:10.1007/BF01135693
1060:, pp. 26â41,
881:10.1007/BFb0081746
855:Shephard's problem
824: †â in
723:
703:
673:
511:
479:
448:
418:
256:
228:
198:
183:Shephard's problem
133:
106:
59:
36:
890:978-3-540-19353-1
807:Koldobsky (1998b)
795:Koldobsky (1998a)
738:intersection body
726:{\displaystyle n}
514:{\displaystyle K}
259:{\displaystyle K}
201:{\displaystyle K}
167:Hermann Minkowski
143:in the direction
62:{\displaystyle K}
1146:
1125:
1095:
1093:
1068:
1052:
1032:
1015:
989:
972:
949:
924:
901:
840: â„ 5.
819:
818:
732:
730:
729:
724:
712:
710:
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704:
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682:
680:
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674:
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388:
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349:
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257:
237:
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142:
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134:
118:support function
115:
113:
112:
107:
105:
104:
68:
66:
65:
60:
45:
43:
42:
37:
1154:
1153:
1149:
1148:
1147:
1145:
1144:
1143:
1139:Convex geometry
1129:
1128:
1098:
1071:
1055:
1035:
992:
952:
922:10.1.1.610.5349
904:
891:
873:Springer-Verlag
866:
863:
846:
817:
812:
811:
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715:
714:
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248:
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215:
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122:
121:
90:
79:
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75:Euclidean space
51:
50:
25:
24:
22:projection body
18:convex geometry
12:
11:
5:
1152:
1150:
1142:
1141:
1131:
1130:
1127:
1126:
1109:(4): 213â222,
1096:
1069:
1053:
1033:
1006:(2): 232â261,
990:
950:
915:(4): 827â840,
902:
889:
862:
859:
858:
857:
852:
845:
842:
813:
777:Funk transform
722:
700:
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684:
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672:
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315:
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305:
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291:
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283:
279:
255:
227:
222:
218:
197:
159:orthogonal to
132:
129:
103:
100:
97:
93:
89:
86:
73:-dimensional
58:
35:
32:
13:
10:
9:
6:
4:
3:
2:
1151:
1140:
1137:
1136:
1134:
1124:
1120:
1116:
1112:
1108:
1104:
1103:
1097:
1092:
1087:
1083:
1080:(in German).
1079:
1075:
1070:
1067:
1063:
1059:
1054:
1051:
1047:
1043:
1039:
1034:
1031:
1027:
1023:
1019:
1014:
1009:
1005:
1001:
1000:
995:
994:Lutwak, Erwin
991:
988:
984:
980:
976:
971:
966:
962:
958:
957:
951:
948:
944:
940:
936:
932:
928:
923:
918:
914:
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909:
903:
900:
896:
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886:
882:
878:
874:
870:
865:
864:
860:
856:
853:
851:
848:
847:
843:
841:
839:
835:
831:
827:
823:
816:
808:
804:
800:
796:
792:
790:
786:
782:
778:
774:
770:
766:
762:
758:
754:
750:
746:
742:
739:
734:
720:
698:
694:
670:
662:
658:
652:
639:
635:
629:
626:
623:
613:
609:
600:
596:
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586:
581:
568:
564:
558:
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552:
544:
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508:
500:
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492:
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443:
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384:
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358:
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331:
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130:
119:
101:
98:
95:
91:
87:
84:
76:
72:
56:
49:
33:
23:
19:
1106:
1100:
1081:
1077:
1057:
1041:
1003:
997:
960:
954:
912:
906:
868:
837:
833:
825:
821:
814:
802:
798:
793:
780:
772:
768:
764:
760:
756:
752:
748:
744:
740:
737:
735:
713:denotes any
685:
490:
459:
458:denotes the
430:
244:Petty (1971)
187:
171:Petty (1967)
165:
160:
152:
148:
144:
70:
21:
15:
963:(1): 1â14,
238:denote the
48:convex body
861:References
240:polar body
157:hyperplane
1084:: 71â82.
1022:0001-8708
979:0001-8708
939:0002-9327
917:CiteSeerX
653:∘
649:Π
627:−
593:≥
582:∘
578:Π
556:−
398:∘
394:Π
372:−
338:≤
327:∘
323:Π
301:−
221:∘
217:Π
175:Schneider
155:onto the
128:Π
99:−
88:∈
31:Π
1133:Category
844:See also
759:is the (
147:is the (
1123:1119653
1066:0362057
1050:0216369
1030:0963487
987:1623669
947:1637955
899:0950986
787: (
775:is the
497: (
177: (
1121:
1064:
1048:
1028:
1020:
985:
977:
945:
937:
919:
897:
887:
830:l norm
785:Lutwak
686:where
431:where
116:, the
20:, the
495:Zhang
46:of a
1018:ISSN
975:ISSN
935:ISSN
885:ISBN
789:1988
736:The
499:1991
188:For
179:1967
173:and
1111:doi
1086:doi
1082:101
1008:doi
965:doi
961:136
927:doi
913:120
877:doi
791:).
743:of
489:is
163:.
120:of
69:in
16:In
1135::
1119:MR
1117:,
1107:39
1105:,
1076:.
1062:MR
1046:MR
1040:,
1026:MR
1024:,
1016:,
1004:71
1002:,
983:MR
981:,
973:,
959:,
943:MR
941:,
933:,
925:,
911:,
895:MR
893:,
883:,
773:IK
753:IK
741:IK
521:,
266:,
185:.
1113::
1094:.
1088::
1010::
967::
929::
879::
838:n
834:n
826:n
822:p
815:n
803:x
799:x
781:K
769:u
765:K
761:n
757:u
749:u
745:K
721:n
699:n
695:T
671:,
668:)
663:n
659:T
645:(
640:n
636:V
630:1
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614:n
610:T
606:(
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597:V
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587:K
574:(
569:n
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537:n
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509:K
491:n
475:n
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444:n
440:B
416:,
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390:(
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351:(
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332:K
319:(
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287:(
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226:K
196:K
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131:K
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57:K
34:K
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