415:
152:
3036:
Set-theoretic, algebraic and topological operations on set-valued functions (like union, composition, sum, convex hull, closure) usually preserve the type of continuity. But this should be taken with appropriate care since, for example, there exists a pair of lower hemicontinuous set-valued functions
3037:
whose intersection is not lower hemicontinuous. This can be fixed upon strengthening continuity properties: if one of those lower hemicontinuous multifunctions has open graph then their intersection is again lower hemicontinuous.
3053:, Bressan–Colombo directionally continuous selection theorem, Fryszkowski decomposable map selection). Likewise, upper hemicontinuous maps admit approximations (e.g. Ancel–Granas–Górniewicz–Kryszewski theorem).
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and approximations to set-valued functions. Typically lower hemicontinuous set-valued functions admit single-valued selections (
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If a set-valued function is both upper hemicontinuous and lower hemicontinuous, it is said to be continuous.
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481:. In contrast, the function is upper hemicontinuous everywhere. For example, considering any sequence
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Crucial to set-valued analysis (in view of applications) are the investigation of single-valued
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548:. In contrast, that function is lower hemicontinuous everywhere. For example, for any sequence
256:
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418:
This set-valued function is lower hemicontinuous everywhere, but not upper hemicontinuous at
155:
This set-valued function is upper hemicontinuous everywhere, but not lower hemicontinuous at
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is an upper hemicontinuous set-valued function with closed domain (that is, the domain of
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17:
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3218:- a theorem about constructing a single-valued function from a set-valued function.
58:. A set-valued function that is both upper and lower hemicontinuous is said to be
3404:
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3352:
3316:
3242:
3878:
3873:
3757:
31:
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62:
in an analogy to the property of the same name for single-valued functions.
3160:
445:
The image on the right shows a function that is not lower hemicontinuous at
3289:
3272:
504:
The image on the left shows a function that is not upper hemicontinuous at
2952:
is a set-valued function with convex values and open upper sections, then
3810:
3061:
The upper and lower hemicontinuity might be viewed as usual continuity:
3591:
1633:
As an example, look at the image at the right, and consider sequence
564:) contains a single point, and there exists a corresponding sequence
477:
is contained in the bottom horizontal line, so it cannot converge to
497:
is contained in the vertical line that is the image of the limit of
1778:{\displaystyle Gr(\Gamma )=\{(a,b)\in A\times B:b\in \Gamma (a)\}.}
520:
contains vertical lines, so there exists a corresponding sequence
413:
150:
3321:(Third ed.). Berlin: Springer Science & Business Media.
489:
from the left or from the right, and any corresponding sequence
118:
Lower hemicontinuity requires that, for any convergent sequence
95:
Upper hemicontinuity requires that, for any convergent sequence
3478:
2433:{\displaystyle b_{\bullet }=\left(b_{k}\right)_{k=1}^{\infty }}
2169:{\displaystyle a_{\bullet }=\left(a_{m}\right)_{m=1}^{\infty }}
1333:{\displaystyle a_{\bullet }=\left(a_{m}\right)_{m=1}^{\infty }}
3354:
Differential
Inclusions: Set-Valued Maps and Viability Theory
3209: – Property of functions which is weaker than continuity
1641:(either from the left or from the right). Then, any sequence
3357:. Grundl. der Math. Wiss. Vol. 264. Berlin: Springer.
2945:{\displaystyle \Gamma :A\to P\left(\mathbb {R} ^{n}\right)}
1645:
that satisfies the requirements converges to some point in
2669:{\displaystyle \Gamma ^{-1}(b)=\{a\in A:b\in \Gamma (a)\}}
2890:
has open lower sections then it is lower hemicontinuous.
3273:"On the Existence of Equilibrium for Abstract Economies"
87:
is contained in the image of the corresponding point in
3441:. New York: Oxford University Press. pp. 949–951.
3211:
Pages displaying short descriptions of redirect targets
2524:{\displaystyle b_{k}\in \Gamma \left(a_{m_{k}}\right)}
2340:{\displaystyle \left(a_{m_{k}}\right)_{k=1}^{\infty }}
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is lower hemicontinuous if and only if the mapping
3897:
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3180:
lower semicontinuous maps in the sense of
Hausdorff
1460:{\displaystyle b_{m}\in \Gamma \left(a_{m}\right),}
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1406:{\displaystyle \left(b_{m}\right)_{m=1}^{\infty }}
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3277:Journal of Mathematical Analysis and Applications
3203: – Distance between two metric-space subsets
3460:. Princeton University Press. pp. 216–226.
1527:
1478:
3490:
2843:{\displaystyle \operatorname {Gr} (\Gamma ),}
461:is a vertical line that contains some point (
99:in a domain, and for any convergent sequence
65:To explain both notions, consider a sequence
8:
3184:metrically lower / upper semicontinuous maps
2663:
2630:
2077:{\displaystyle \Gamma :A\rightrightarrows B}
2032:is compact, then the converse is also true.
2002:{\displaystyle \operatorname {Gr} (\Gamma )}
1895:{\displaystyle \Gamma :A\rightrightarrows B}
1769:
1718:
1686:{\displaystyle \Gamma :A\rightrightarrows B}
1629:is compact, then the converse is also true.
1206:{\displaystyle \Gamma :A\rightrightarrows B}
867:{\displaystyle \Gamma :A\rightrightarrows B}
614:{\displaystyle \Gamma :A\rightrightarrows B}
524:in which all elements are bounded away from
3159:(For the notion of hyperspace compare also
3065:
2894:
2047:
1864:
1561:{\displaystyle \lim _{m\to \infty }b_{m}=b}
1512:{\displaystyle \lim _{m\to \infty }a_{m}=a}
1175:
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2870:has open upper and lower sections and if
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3458:Real Analysis with Economic Applications
3000:{\displaystyle A\times \mathbb {R} ^{n}}
1141:{\displaystyle V\cap S\neq \varnothing }
3587:Locally convex topological vector space
3228:
1135:
544:), so it does not contain the limit of
27:Semicontinuity for set-valued functions
441:because the graph (set) is not closed.
69:of points in a domain, and a sequence
134:that corresponds to a subsequence of
7:
3032:Operations Preserving Hemicontinuity
708:{\displaystyle \Gamma (a)\subset V,}
404:{\displaystyle f\left(x_{m}\right).}
73:of points in the range. We say that
1922:is closed) and closed values (i.e.
1661:The graph of a set-valued function
556:, from the left or from the right,
3406:Multivalued Differential Equations
3110:
3078:
3014:
2959:
2907:
2877:
2857:
2831:
2802:
2775:
2732:
2651:
2606:
2569:
2493:
2425:
2332:
2262:
2161:
2110:if and only if for every sequence
2059:
1993:
1929:
1909:
1877:
1838:
1792:
1757:
1709:
1668:
1581:
1537:
1488:
1433:
1398:
1325:
1220:
1188:
1003:
931:
849:
790:
684:
596:
316:{\displaystyle \left(y_{m}\right)}
203:{\displaystyle \left(x_{m}\right)}
25:
3174:we can also define the so-called
3151:has been endowed with the lower
3137:{\displaystyle \Gamma :A\to P(B)}
2466:{\displaystyle b_{\bullet }\to b}
2222:{\displaystyle a_{\bullet }\to a}
175: : for a sequence of points
42:are extensions of the notions of
3170:Using lower and upper Hausdorff
2280:{\displaystyle b\in \Gamma (a),}
1637:in the domain that converges to
1599:{\displaystyle b\in \Gamma (a).}
512:be a sequence that converges to
453:be a sequence that converges to
3692:Ekeland's variational principle
122:in a domain, and for any point
3926:Theory of continuous functions
3131:
3125:
3119:
3096:{\displaystyle \Gamma :A\to B}
3087:
2916:
2834:
2828:
2660:
2654:
2624:
2618:
2587:{\displaystyle \Gamma :A\to B}
2578:
2457:
2271:
2265:
2213:
2068:
1996:
1990:
1938:
1932:
1886:
1847:
1841:
1766:
1760:
1733:
1721:
1712:
1706:
1677:
1590:
1584:
1534:
1485:
1238:{\displaystyle \Gamma :A\to B}
1229:
1197:
1012:
1006:
940:
934:
858:
799:
793:
693:
687:
605:
275:
269:
1:
956:there exists a neighbourhood
715:there exists a neighbourhood
532:). The image of the limit of
516:from the right. The image of
126:in the image of the limit of
3057:Other concepts of continuity
2745:values are all open sets in
2367:{\displaystyle a_{\bullet }}
1116:means nonempty intersection
457:from the left. The image of
107:, the image of the limit of
3712:Hermite–Hadamard inequality
2287:there exists a subsequence
2084:is lower hemicontinuous at
2042:Sequential characterization
1245:is upper hemicontinuous at
1170:Sequential characterization
949:{\displaystyle \Gamma (a),}
288:) such that no sequence of
3952:
3309:Aliprantis, Charalambos D.
3271:Zhou, J.X. (August 1995).
1944:{\displaystyle \Gamma (a)}
1853:{\displaystyle \Gamma (a)}
1181:For a set-valued function
1018:{\displaystyle \Gamma (x)}
805:{\displaystyle \Gamma (x)}
670:{\displaystyle V\subset B}
130:, there exists a sequence
3051:Michael selection theorem
3027:is lower hemicontinuous.
281:{\displaystyle y\in f(x)}
3898:Applications and related
3702:Fenchel-Young inequality
3144:is continuous where the
1274:then for every sequence
536:contains a single point
3658:Legendre transformation
3582:Legendre transformation
3020:{\displaystyle \Gamma }
2965:{\displaystyle \Gamma }
2883:{\displaystyle \Gamma }
2863:{\displaystyle \Gamma }
2808:{\displaystyle \Gamma }
2781:{\displaystyle \Gamma }
2738:{\displaystyle \Gamma }
2718:{\displaystyle b\in B.}
1915:{\displaystyle \Gamma }
1798:{\displaystyle \Gamma }
1267:{\displaystyle a\in A,}
1213:with closed values, if
1067:{\displaystyle x\in U.}
777:{\displaystyle x\in U,}
3905:Convexity in economics
3839:(lower) ideally convex
3697:Fenchel–Moreau theorem
3687:Carathéodory's theorem
3438:Microeconomic Analysis
3290:10.1006/jmaa.1995.1271
3196:Differential inclusion
3138:
3097:
3021:
3001:
2966:
2946:
2884:
2864:
2844:
2809:
2782:
2762:
2739:
2719:
2690:
2670:
2588:
2562:A set-valued function
2548:
2525:
2467:
2434:
2368:
2341:
2281:
2243:
2223:
2190:
2170:
2104:
2103:{\displaystyle a\in A}
2078:
2026:
2003:
1971:
1970:{\displaystyle a\in A}
1945:
1916:
1896:
1854:
1825:
1824:{\displaystyle a\in A}
1799:
1779:
1693:is the set defined by
1687:
1623:
1600:
1562:
1513:
1461:
1407:
1354:
1334:
1268:
1239:
1207:
1142:
1110:
1088:
1068:
1039:
1019:
990:
970:
950:
918:
904:if for every open set
898:
897:{\displaystyle a\in A}
868:
842:A set-valued function
829:
806:
778:
749:
729:
709:
671:
645:
644:{\displaystyle a\in A}
615:
589:A set-valued function
469:). But every sequence
442:
435:
411:
405:
364:
337:
317:
282:
247:
227:
204:
169:
111:contains the limit of
3931:Mathematical analysis
3827:Convex series related
3727:Shapley–Folkman lemma
3409:. Walter de Gruyter.
3385:. Basel: Birkhäuser.
3247:. Basel: Birkhäuser.
3235:Proposition 1.4.8 of
3139:
3098:
3022:
3002:
2972:has an open graph in
2967:
2947:
2885:
2865:
2845:
2810:
2783:
2763:
2740:
2720:
2691:
2671:
2589:
2549:
2526:
2468:
2435:
2369:
2342:
2282:
2244:
2224:
2191:
2171:
2105:
2079:
2027:
2004:
1972:
1946:
1917:
1897:
1855:
1826:
1800:
1780:
1688:
1624:
1601:
1563:
1514:
1462:
1408:
1355:
1335:
1269:
1240:
1208:
1143:
1111:
1089:
1069:
1040:
1020:
991:
971:
951:
919:
899:
869:
830:
807:
779:
750:
730:
710:
672:
646:
616:
436:
417:
406:
365:
363:{\displaystyle y_{m}}
338:
318:
283:
248:
228:
205:
170:
154:
3936:Variational analysis
3717:Krein–Milman theorem
3510:variational analysis
3429:Whinston, Michael D.
3237:Aubin, Jean-Pierre;
3107:
3075:
3011:
2976:
2956:
2904:
2874:
2854:
2819:
2799:
2772:
2749:
2729:
2700:
2680:
2602:
2566:
2535:
2477:
2444:
2378:
2374:and also a sequence
2351:
2291:
2253:
2233:
2200:
2180:
2114:
2088:
2056:
2037:Lower hemicontinuity
2016:
1981:
1955:
1926:
1906:
1874:
1835:
1809:
1789:
1697:
1665:
1657:Closed graph theorem
1613:
1572:
1523:
1474:
1417:
1364:
1344:
1278:
1249:
1217:
1185:
1165:Upper hemicontinuity
1120:
1100:
1078:
1049:
1029:
1000:
980:
960:
928:
908:
882:
876:lower hemicontinuous
846:
838:Lower hemicontinuity
816:
787:
759:
739:
719:
681:
655:
629:
623:upper hemicontinuous
593:
585:Upper hemicontinuity
473:that corresponds to
422:
374:
347:
327:
292:
257:
237:
214:
179:
159:
138:, that converges to
103:that corresponds to
56:set-valued functions
48:lower semicontinuity
40:lower hemicontinuity
36:upper hemicontinuity
18:Lower hemicontinuous
3707:Jensen's inequality
3577:Lagrange multiplier
3567:Convex optimization
3562:Convex metric space
3456:Ok, Efe A. (2007).
3382:Set-Valued Analysis
3244:Set-Valued Analysis
3069: —
3041:Function Selections
2898: —
2790:open upper sections
2596:open lower sections
2429:
2336:
2165:
2051: —
1868: —
1402:
1360:and every sequence
1329:
1179: —
651:if, for every open
508:. To see this, let
449:. To see this, let
3835:(cs, bcs)-complete
3806:Algebraic interior
3524:Convex combination
3425:Mas-Colell, Andreu
3377:Frankowska, Hélène
3373:Aubin, Jean-Pierre
3345:Aubin, Jean-Pierre
3239:Frankowska, Hélène
3201:Hausdorff distance
3134:
3093:
3067:
3017:
2997:
2962:
2942:
2896:
2895:Open Graph Theorem
2880:
2860:
2840:
2815:has an open graph
2805:
2778:
2761:{\displaystyle B,}
2758:
2735:
2715:
2686:
2666:
2584:
2558:Open graph theorem
2547:{\displaystyle k.}
2544:
2521:
2463:
2430:
2394:
2364:
2337:
2294:
2277:
2239:
2219:
2186:
2166:
2130:
2100:
2074:
2049:
2022:
1999:
1967:
1951:is closed for all
1941:
1912:
1892:
1866:
1850:
1821:
1805:is the set of all
1795:
1775:
1683:
1619:
1596:
1558:
1541:
1509:
1492:
1457:
1403:
1367:
1350:
1330:
1294:
1264:
1235:
1203:
1177:
1138:
1106:
1084:
1064:
1035:
1015:
986:
966:
946:
914:
894:
864:
828:{\displaystyle V.}
825:
802:
774:
755:such that for all
745:
725:
705:
667:
641:
611:
568:that converges to
552:that converges to
485:that converges to
443:
434:{\displaystyle x,}
431:
412:
401:
360:
333:
313:
278:
243:
226:{\displaystyle x,}
223:
210:that converges to
200:
165:
3913:
3912:
3467:978-0-691-11768-3
3328:978-3-540-29587-7
3216:Selection theorem
3153:Vietoris topology
3071:A set-valued map
2689:{\displaystyle A}
2242:{\displaystyle A}
2189:{\displaystyle A}
2025:{\displaystyle B}
1622:{\displaystyle B}
1526:
1477:
1353:{\displaystyle A}
1109:{\displaystyle S}
1087:{\displaystyle V}
1038:{\displaystyle V}
989:{\displaystyle a}
969:{\displaystyle U}
917:{\displaystyle V}
748:{\displaystyle a}
728:{\displaystyle U}
336:{\displaystyle y}
246:{\displaystyle y}
168:{\displaystyle x}
83:if each point in
50:of single-valued
16:(Redirected from
3943:
3831:(cs, lcs)-closed
3777:Effective domain
3732:Robinson–Ursescu
3608:Convex conjugate
3499:
3492:
3485:
3476:
3471:
3452:
3420:
3396:
3368:
3340:
3295:
3294:
3292:
3268:
3259:
3258:
3233:
3212:
3143:
3141:
3140:
3135:
3102:
3100:
3099:
3094:
3070:
3026:
3024:
3023:
3018:
3006:
3004:
3003:
2998:
2996:
2995:
2990:
2971:
2969:
2968:
2963:
2951:
2949:
2948:
2943:
2941:
2937:
2936:
2931:
2899:
2889:
2887:
2886:
2881:
2869:
2867:
2866:
2861:
2849:
2847:
2846:
2841:
2814:
2812:
2811:
2806:
2788:is said to have
2787:
2785:
2784:
2779:
2767:
2765:
2764:
2759:
2744:
2742:
2741:
2736:
2724:
2722:
2721:
2716:
2695:
2693:
2692:
2687:
2675:
2673:
2672:
2667:
2617:
2616:
2594:is said to have
2593:
2591:
2590:
2585:
2553:
2551:
2550:
2545:
2530:
2528:
2527:
2522:
2520:
2516:
2515:
2514:
2513:
2489:
2488:
2472:
2470:
2469:
2464:
2456:
2455:
2439:
2437:
2436:
2431:
2428:
2423:
2412:
2408:
2407:
2390:
2389:
2373:
2371:
2370:
2365:
2363:
2362:
2346:
2344:
2343:
2338:
2335:
2330:
2319:
2315:
2314:
2313:
2312:
2286:
2284:
2283:
2278:
2248:
2246:
2245:
2240:
2228:
2226:
2225:
2220:
2212:
2211:
2195:
2193:
2192:
2187:
2175:
2173:
2172:
2167:
2164:
2159:
2148:
2144:
2143:
2126:
2125:
2109:
2107:
2106:
2101:
2083:
2081:
2080:
2075:
2052:
2031:
2029:
2028:
2023:
2008:
2006:
2005:
2000:
1976:
1974:
1973:
1968:
1950:
1948:
1947:
1942:
1921:
1919:
1918:
1913:
1901:
1899:
1898:
1893:
1869:
1859:
1857:
1856:
1851:
1830:
1828:
1827:
1822:
1804:
1802:
1801:
1796:
1784:
1782:
1781:
1776:
1692:
1690:
1689:
1684:
1628:
1626:
1625:
1620:
1605:
1603:
1602:
1597:
1567:
1565:
1564:
1559:
1551:
1550:
1540:
1518:
1516:
1515:
1510:
1502:
1501:
1491:
1466:
1464:
1463:
1458:
1453:
1449:
1448:
1429:
1428:
1412:
1410:
1409:
1404:
1401:
1396:
1385:
1381:
1380:
1359:
1357:
1356:
1351:
1339:
1337:
1336:
1331:
1328:
1323:
1312:
1308:
1307:
1290:
1289:
1273:
1271:
1270:
1265:
1244:
1242:
1241:
1236:
1212:
1210:
1209:
1204:
1180:
1147:
1145:
1144:
1139:
1115:
1113:
1112:
1107:
1093:
1091:
1090:
1085:
1073:
1071:
1070:
1065:
1044:
1042:
1041:
1036:
1024:
1022:
1021:
1016:
995:
993:
992:
987:
975:
973:
972:
967:
955:
953:
952:
947:
923:
921:
920:
915:
903:
901:
900:
895:
873:
871:
870:
865:
834:
832:
831:
826:
811:
809:
808:
803:
783:
781:
780:
775:
754:
752:
751:
746:
734:
732:
731:
726:
714:
712:
711:
706:
676:
674:
673:
668:
650:
648:
647:
642:
620:
618:
617:
612:
440:
438:
437:
432:
410:
408:
407:
402:
397:
393:
392:
369:
367:
366:
361:
359:
358:
342:
340:
339:
334:
322:
320:
319:
314:
312:
308:
307:
287:
285:
284:
279:
252:
250:
249:
244:
232:
230:
229:
224:
209:
207:
206:
201:
199:
195:
194:
174:
172:
171:
166:
21:
3951:
3950:
3946:
3945:
3944:
3942:
3941:
3940:
3916:
3915:
3914:
3909:
3893:
3860:
3815:
3746:
3672:
3663:Semi-continuity
3648:Convex function
3629:Logarithmically
3596:
3557:Convex geometry
3538:
3529:Convex function
3512:
3506:Convex analysis
3503:
3468:
3455:
3449:
3433:Green, Jerry R.
3423:
3417:
3401:Deimling, Klaus
3399:
3393:
3371:
3365:
3349:Cellina, Arrigo
3343:
3329:
3307:
3304:
3299:
3298:
3270:
3269:
3262:
3255:
3236:
3234:
3230:
3225:
3210:
3192:
3182:(also known as
3157:
3105:
3104:
3073:
3072:
3068:
3059:
3043:
3034:
3029:
3009:
3008:
3007:if and only if
2985:
2974:
2973:
2954:
2953:
2926:
2922:
2902:
2901:
2897:
2872:
2871:
2852:
2851:
2817:
2816:
2797:
2796:
2770:
2769:
2747:
2746:
2727:
2726:
2698:
2697:
2678:
2677:
2605:
2600:
2599:
2564:
2563:
2560:
2555:
2533:
2532:
2505:
2500:
2496:
2480:
2475:
2474:
2447:
2442:
2441:
2399:
2395:
2381:
2376:
2375:
2354:
2349:
2348:
2304:
2299:
2295:
2289:
2288:
2251:
2250:
2231:
2230:
2203:
2198:
2197:
2178:
2177:
2135:
2131:
2117:
2112:
2111:
2086:
2085:
2054:
2053:
2050:
2044:
2039:
2034:
2014:
2013:
1979:
1978:
1953:
1952:
1924:
1923:
1904:
1903:
1872:
1871:
1867:
1833:
1832:
1807:
1806:
1787:
1786:
1695:
1694:
1663:
1662:
1659:
1631:
1611:
1610:
1570:
1569:
1542:
1521:
1520:
1493:
1472:
1471:
1440:
1436:
1420:
1415:
1414:
1372:
1368:
1362:
1361:
1342:
1341:
1299:
1295:
1281:
1276:
1275:
1247:
1246:
1215:
1214:
1183:
1182:
1178:
1172:
1167:
1162:
1154:
1118:
1117:
1098:
1097:
1076:
1075:
1047:
1046:
1027:
1026:
998:
997:
978:
977:
958:
957:
926:
925:
906:
905:
880:
879:
844:
843:
840:
814:
813:
812:is a subset of
785:
784:
757:
756:
737:
736:
717:
716:
679:
678:
653:
652:
627:
626:
591:
590:
587:
582:
493:, the limit of
420:
419:
384:
380:
372:
371:
350:
345:
344:
325:
324:
299:
295:
290:
289:
255:
254:
235:
234:
212:
211:
186:
182:
177:
176:
157:
156:
149:
78:corresponds to
28:
23:
22:
15:
12:
11:
5:
3949:
3947:
3939:
3938:
3933:
3928:
3918:
3917:
3911:
3910:
3908:
3907:
3901:
3899:
3895:
3894:
3892:
3891:
3886:
3884:Strong duality
3881:
3876:
3870:
3868:
3862:
3861:
3859:
3858:
3823:
3821:
3817:
3816:
3814:
3813:
3808:
3799:
3794:
3792:John ellipsoid
3789:
3784:
3779:
3774:
3760:
3754:
3752:
3748:
3747:
3745:
3744:
3739:
3734:
3729:
3724:
3719:
3714:
3709:
3704:
3699:
3694:
3689:
3683:
3681:
3679:results (list)
3674:
3673:
3671:
3670:
3665:
3660:
3655:
3653:Invex function
3650:
3641:
3636:
3631:
3626:
3621:
3615:
3610:
3604:
3602:
3598:
3597:
3595:
3594:
3589:
3584:
3579:
3574:
3569:
3564:
3559:
3554:
3552:Choquet theory
3548:
3546:
3540:
3539:
3537:
3536:
3531:
3526:
3520:
3518:
3517:Basic concepts
3514:
3513:
3504:
3502:
3501:
3494:
3487:
3479:
3473:
3472:
3466:
3453:
3447:
3421:
3415:
3397:
3391:
3369:
3363:
3341:
3327:
3313:Border, Kim C.
3303:
3300:
3297:
3296:
3283:(3): 839–858.
3260:
3253:
3227:
3226:
3224:
3221:
3220:
3219:
3213:
3207:Semicontinuity
3204:
3198:
3191:
3188:
3165:function space
3133:
3130:
3127:
3124:
3121:
3118:
3115:
3112:
3092:
3089:
3086:
3083:
3080:
3063:
3058:
3055:
3042:
3039:
3033:
3030:
3016:
2994:
2989:
2984:
2981:
2961:
2940:
2935:
2930:
2925:
2921:
2918:
2915:
2912:
2909:
2892:
2879:
2859:
2839:
2836:
2833:
2830:
2827:
2824:
2804:
2791:
2777:
2757:
2754:
2734:
2714:
2711:
2708:
2705:
2685:
2665:
2662:
2659:
2656:
2653:
2650:
2647:
2644:
2641:
2638:
2635:
2632:
2629:
2626:
2623:
2620:
2615:
2612:
2608:
2597:
2583:
2580:
2577:
2574:
2571:
2559:
2556:
2543:
2540:
2519:
2512:
2508:
2503:
2499:
2495:
2492:
2487:
2483:
2462:
2459:
2454:
2450:
2427:
2422:
2419:
2416:
2411:
2406:
2402:
2398:
2393:
2388:
2384:
2361:
2357:
2334:
2329:
2326:
2323:
2318:
2311:
2307:
2302:
2298:
2276:
2273:
2270:
2267:
2264:
2261:
2258:
2238:
2218:
2215:
2210:
2206:
2185:
2163:
2158:
2155:
2152:
2147:
2142:
2138:
2134:
2129:
2124:
2120:
2099:
2096:
2093:
2073:
2070:
2067:
2064:
2061:
2045:
2043:
2040:
2038:
2035:
2021:
1998:
1995:
1992:
1989:
1986:
1966:
1963:
1960:
1940:
1937:
1934:
1931:
1911:
1891:
1888:
1885:
1882:
1879:
1862:
1860:is not empty.
1849:
1846:
1843:
1840:
1820:
1817:
1814:
1794:
1774:
1771:
1768:
1765:
1762:
1759:
1756:
1753:
1750:
1747:
1744:
1741:
1738:
1735:
1732:
1729:
1726:
1723:
1720:
1717:
1714:
1711:
1708:
1705:
1702:
1682:
1679:
1676:
1673:
1670:
1658:
1655:
1618:
1607:
1606:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1557:
1554:
1549:
1545:
1539:
1536:
1533:
1529:
1508:
1505:
1500:
1496:
1490:
1487:
1484:
1480:
1456:
1452:
1447:
1443:
1439:
1435:
1432:
1427:
1423:
1400:
1395:
1392:
1389:
1384:
1379:
1375:
1371:
1349:
1327:
1322:
1319:
1316:
1311:
1306:
1302:
1298:
1293:
1288:
1284:
1263:
1260:
1257:
1254:
1234:
1231:
1228:
1225:
1222:
1202:
1199:
1196:
1193:
1190:
1173:
1171:
1168:
1166:
1163:
1161:
1158:
1153:
1150:
1137:
1134:
1131:
1128:
1125:
1105:
1096:
1083:
1063:
1060:
1057:
1054:
1034:
1014:
1011:
1008:
1005:
985:
965:
945:
942:
939:
936:
933:
913:
893:
890:
887:
874:is said to be
863:
860:
857:
854:
851:
839:
836:
824:
821:
801:
798:
795:
792:
773:
770:
767:
764:
744:
724:
704:
701:
698:
695:
692:
689:
686:
666:
663:
660:
640:
637:
634:
621:is said to be
610:
607:
604:
601:
598:
586:
583:
581:
578:
430:
427:
400:
396:
391:
387:
383:
379:
357:
353:
332:
311:
306:
302:
298:
277:
274:
271:
268:
265:
262:
242:
222:
219:
198:
193:
189:
185:
164:
148:
145:
144:
143:
116:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3948:
3937:
3934:
3932:
3929:
3927:
3924:
3923:
3921:
3906:
3903:
3902:
3900:
3896:
3890:
3887:
3885:
3882:
3880:
3877:
3875:
3872:
3871:
3869:
3867:
3863:
3856:
3854:
3848:
3846:
3840:
3836:
3832:
3828:
3825:
3824:
3822:
3818:
3812:
3809:
3807:
3803:
3800:
3798:
3795:
3793:
3790:
3788:
3785:
3783:
3780:
3778:
3775:
3773:
3769:
3765:
3761:
3759:
3756:
3755:
3753:
3749:
3743:
3740:
3738:
3735:
3733:
3730:
3728:
3725:
3723:
3722:Mazur's lemma
3720:
3718:
3715:
3713:
3710:
3708:
3705:
3703:
3700:
3698:
3695:
3693:
3690:
3688:
3685:
3684:
3682:
3680:
3675:
3669:
3668:Subderivative
3666:
3664:
3661:
3659:
3656:
3654:
3651:
3649:
3645:
3642:
3640:
3637:
3635:
3632:
3630:
3627:
3625:
3622:
3620:
3616:
3614:
3611:
3609:
3606:
3605:
3603:
3599:
3593:
3590:
3588:
3585:
3583:
3580:
3578:
3575:
3573:
3570:
3568:
3565:
3563:
3560:
3558:
3555:
3553:
3550:
3549:
3547:
3545:
3544:Topics (list)
3541:
3535:
3532:
3530:
3527:
3525:
3522:
3521:
3519:
3515:
3511:
3507:
3500:
3495:
3493:
3488:
3486:
3481:
3480:
3477:
3469:
3463:
3459:
3454:
3450:
3448:0-19-507340-1
3444:
3440:
3439:
3434:
3430:
3426:
3422:
3418:
3416:3-11-013212-5
3412:
3408:
3407:
3402:
3398:
3394:
3392:3-7643-3478-9
3388:
3384:
3383:
3378:
3374:
3370:
3366:
3364:0-387-13105-1
3360:
3356:
3355:
3350:
3346:
3342:
3338:
3334:
3330:
3324:
3320:
3319:
3314:
3310:
3306:
3305:
3301:
3291:
3286:
3282:
3278:
3274:
3267:
3265:
3261:
3256:
3254:3-7643-3478-9
3250:
3246:
3245:
3240:
3232:
3229:
3222:
3217:
3214:
3208:
3205:
3202:
3199:
3197:
3194:
3193:
3189:
3187:
3185:
3181:
3177:
3173:
3168:
3166:
3162:
3156:
3154:
3150:
3147:
3128:
3122:
3116:
3113:
3090:
3084:
3081:
3062:
3056:
3054:
3052:
3048:
3040:
3038:
3031:
3028:
2992:
2982:
2979:
2938:
2933:
2923:
2919:
2913:
2910:
2891:
2837:
2825:
2822:
2793:
2789:
2755:
2752:
2712:
2709:
2706:
2703:
2683:
2657:
2648:
2645:
2642:
2639:
2636:
2633:
2627:
2621:
2613:
2610:
2595:
2581:
2575:
2572:
2557:
2554:
2541:
2538:
2517:
2510:
2506:
2501:
2497:
2490:
2485:
2481:
2460:
2452:
2448:
2420:
2417:
2414:
2409:
2404:
2400:
2396:
2391:
2386:
2382:
2359:
2355:
2327:
2324:
2321:
2316:
2309:
2305:
2300:
2296:
2274:
2268:
2259:
2256:
2236:
2216:
2208:
2204:
2183:
2156:
2153:
2150:
2145:
2140:
2136:
2132:
2127:
2122:
2118:
2097:
2094:
2091:
2071:
2065:
2062:
2041:
2036:
2033:
2019:
2010:
1987:
1984:
1964:
1961:
1958:
1935:
1889:
1883:
1880:
1861:
1844:
1818:
1815:
1812:
1785:The graph of
1772:
1763:
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1730:
1727:
1724:
1715:
1703:
1700:
1680:
1674:
1671:
1656:
1654:
1652:
1648:
1644:
1640:
1636:
1630:
1616:
1593:
1587:
1578:
1575:
1555:
1552:
1547:
1543:
1531:
1506:
1503:
1498:
1494:
1482:
1469:
1468:
1467:
1454:
1450:
1445:
1441:
1437:
1430:
1425:
1421:
1393:
1390:
1387:
1382:
1377:
1373:
1369:
1347:
1320:
1317:
1314:
1309:
1304:
1300:
1296:
1291:
1286:
1282:
1261:
1258:
1255:
1252:
1232:
1226:
1223:
1200:
1194:
1191:
1169:
1164:
1159:
1157:
1151:
1149:
1132:
1129:
1126:
1123:
1103:
1094:
1081:
1061:
1058:
1055:
1052:
1032:
1009:
983:
963:
943:
937:
924:intersecting
911:
891:
888:
885:
878:at the point
877:
861:
855:
852:
837:
835:
822:
819:
796:
771:
768:
765:
762:
742:
722:
702:
699:
696:
690:
664:
661:
658:
638:
635:
632:
624:
608:
602:
599:
584:
579:
577:
575:
571:
567:
563:
559:
555:
551:
547:
543:
539:
535:
531:
527:
523:
519:
515:
511:
507:
502:
500:
496:
492:
488:
484:
480:
476:
472:
468:
464:
460:
456:
452:
448:
428:
425:
416:
398:
394:
389:
385:
381:
377:
355:
351:
330:
323:converges to
309:
304:
300:
296:
272:
266:
263:
260:
240:
220:
217:
196:
191:
187:
183:
162:
153:
146:
141:
137:
133:
129:
125:
121:
117:
114:
110:
106:
102:
98:
94:
93:
92:
90:
86:
82:
81:
77:
72:
68:
63:
61:
57:
53:
49:
45:
41:
37:
33:
19:
3889:Weak duality
3852:
3844:
3764:Orthogonally
3457:
3437:
3405:
3381:
3353:
3317:
3280:
3276:
3243:
3231:
3183:
3179:
3175:
3169:
3158:
3148:
3064:
3060:
3044:
3035:
2893:
2794:
2561:
2046:
2011:
2009:is closed.
1863:
1660:
1650:
1646:
1642:
1638:
1634:
1632:
1608:
1174:
1155:
875:
841:
622:
588:
573:
569:
565:
561:
557:
553:
549:
545:
541:
537:
533:
529:
525:
521:
517:
513:
509:
505:
503:
498:
494:
490:
486:
482:
478:
474:
470:
466:
462:
458:
454:
450:
446:
444:
139:
135:
131:
127:
123:
119:
112:
108:
104:
100:
96:
88:
84:
79:
75:
74:
70:
66:
64:
59:
39:
35:
29:
3879:Duality gap
3874:Dual system
3758:Convex hull
2676:is open in
2598:if the set
1025:intersects
625:at a point
580:Definitions
343:where each
32:mathematics
3920:Categories
3802:Radial set
3772:Convex set
3534:Convex set
3302:References
3172:uniformity
3146:hyperspace
3047:selections
2696:for every
2531:for every
2440:such that
2196:such that
1831:such that
1413:such that
1160:Properties
1152:Continuity
1095:intersects
996:such that
233:we have a
60:continuous
3787:Hypograph
3337:262692874
3161:power set
3120:→
3111:Γ
3088:→
3079:Γ
3015:Γ
2983:×
2960:Γ
2917:→
2908:Γ
2878:Γ
2858:Γ
2832:Γ
2826:
2803:Γ
2776:Γ
2733:Γ
2707:∈
2652:Γ
2649:∈
2637:∈
2611:−
2607:Γ
2579:→
2570:Γ
2494:Γ
2491:∈
2458:→
2453:∙
2426:∞
2387:∙
2360:∙
2333:∞
2263:Γ
2260:∈
2214:→
2209:∙
2162:∞
2123:∙
2095:∈
2069:⇉
2060:Γ
1994:Γ
1988:
1962:∈
1930:Γ
1910:Γ
1887:⇉
1878:Γ
1839:Γ
1816:∈
1793:Γ
1758:Γ
1755:∈
1743:×
1737:∈
1710:Γ
1678:⇉
1669:Γ
1582:Γ
1579:∈
1538:∞
1535:→
1489:∞
1486:→
1434:Γ
1431:∈
1399:∞
1326:∞
1287:∙
1256:∈
1230:→
1221:Γ
1198:⇉
1189:Γ
1136:∅
1133:≠
1127:∩
1056:∈
1004:Γ
932:Γ
889:∈
859:⇉
850:Γ
791:Γ
766:∈
697:⊂
685:Γ
662:⊂
636:∈
606:⇉
597:Γ
264:∈
52:functions
3811:Zonotope
3782:Epigraph
3435:(1995).
3403:(1992).
3379:(1990).
3351:(1984).
3315:(2006).
3241:(1990).
3190:See also
2249:and all
1977:), then
1045:for all
147:Examples
3866:Duality
3768:Pseudo-
3742:Ursescu
3639:Pseudo-
3613:Concave
3592:Simplex
3572:Duality
3066:Theorem
2048:Theorem
1865:Theorem
1176:Theorem
3849:, and
3820:Series
3737:Simons
3644:Quasi-
3634:Proper
3619:Closed
3464:
3445:
3413:
3389:
3361:
3335:
3325:
3251:
1074:(Here
370:is in
3677:Main
3223:Notes
3176:upper
2850:then
2768:then
1568:then
677:with
44:upper
3797:Lens
3751:Sets
3601:Maps
3508:and
3462:ISBN
3443:ISBN
3411:ISBN
3387:ISBN
3359:ISBN
3333:OCLC
3323:ISBN
3249:ISBN
3178:and
3163:and
3149:P(B)
2473:and
1519:and
46:and
38:and
3851:(Hw
3285:doi
3281:193
3186:).
3167:).
2900:If
2795:If
2792:.
2725:If
2347:of
2229:in
2176:in
2012:If
1870:If
1653:).
1609:If
1528:lim
1479:lim
1470:if
1340:in
1148:).
976:of
735:of
576:).
54:to
30:In
3922::
3843:(H
3841:,
3837:,
3833:,
3770:)
3766:,
3646:)
3624:K-
3431:;
3427:;
3375:;
3347:;
3331:.
3311:;
3279:.
3275:.
3263:^
3155:.
2823:Gr
1985:Gr
501:.
91:.
34:,
3857:)
3855:)
3853:x
3847:)
3845:x
3829:(
3804:/
3762:(
3617:(
3498:e
3491:t
3484:v
3470:.
3451:.
3419:.
3395:.
3367:.
3339:.
3293:.
3287::
3257:.
3132:)
3129:B
3126:(
3123:P
3117:A
3114::
3091:B
3085:A
3082::
2993:n
2988:R
2980:A
2939:)
2934:n
2929:R
2924:(
2920:P
2914:A
2911::
2838:,
2835:)
2829:(
2756:,
2753:B
2713:.
2710:B
2704:b
2684:A
2664:}
2661:)
2658:a
2655:(
2646:b
2643::
2640:A
2634:a
2631:{
2628:=
2625:)
2622:b
2619:(
2614:1
2582:B
2576:A
2573::
2542:.
2539:k
2518:)
2511:k
2507:m
2502:a
2498:(
2486:k
2482:b
2461:b
2449:b
2421:1
2418:=
2415:k
2410:)
2405:k
2401:b
2397:(
2392:=
2383:b
2356:a
2328:1
2325:=
2322:k
2317:)
2310:k
2306:m
2301:a
2297:(
2275:,
2272:)
2269:a
2266:(
2257:b
2237:A
2217:a
2205:a
2184:A
2157:1
2154:=
2151:m
2146:)
2141:m
2137:a
2133:(
2128:=
2119:a
2098:A
2092:a
2072:B
2066:A
2063::
2020:B
1997:)
1991:(
1965:A
1959:a
1939:)
1936:a
1933:(
1890:B
1884:A
1881::
1848:)
1845:a
1842:(
1819:A
1813:a
1773:.
1770:}
1767:)
1764:a
1761:(
1752:b
1749::
1746:B
1740:A
1734:)
1731:b
1728:,
1725:a
1722:(
1719:{
1716:=
1713:)
1707:(
1704:r
1701:G
1681:B
1675:A
1672::
1651:x
1649:(
1647:f
1643:b
1639:x
1635:a
1617:B
1594:.
1591:)
1588:a
1585:(
1576:b
1556:b
1553:=
1548:m
1544:b
1532:m
1507:a
1504:=
1499:m
1495:a
1483:m
1455:,
1451:)
1446:m
1442:a
1438:(
1426:m
1422:b
1394:1
1391:=
1388:m
1383:)
1378:m
1374:b
1370:(
1348:A
1321:1
1318:=
1315:m
1310:)
1305:m
1301:a
1297:(
1292:=
1283:a
1262:,
1259:A
1253:a
1233:B
1227:A
1224::
1201:B
1195:A
1192::
1130:S
1124:V
1104:S
1082:V
1062:.
1059:U
1053:x
1033:V
1013:)
1010:x
1007:(
984:a
964:U
944:,
941:)
938:a
935:(
912:V
892:A
886:a
862:B
856:A
853::
823:.
820:V
800:)
797:x
794:(
772:,
769:U
763:x
743:a
723:U
703:,
700:V
694:)
691:a
688:(
665:B
659:V
639:A
633:a
609:B
603:A
600::
574:x
572:(
570:f
566:b
562:x
560:(
558:f
554:x
550:a
546:b
542:x
540:(
538:f
534:a
530:x
528:(
526:f
522:b
518:a
514:x
510:a
506:x
499:a
495:b
491:b
487:x
483:a
479:y
475:a
471:b
467:y
465:,
463:x
459:x
455:x
451:a
447:x
429:,
426:x
399:.
395:)
390:m
386:x
382:(
378:f
356:m
352:y
331:y
310:)
305:m
301:y
297:(
276:)
273:x
270:(
267:f
261:y
253:(
241:y
221:,
218:x
197:)
192:m
188:x
184:(
163:x
142:.
140:x
136:a
132:b
128:a
124:x
120:a
115:.
113:b
109:a
105:a
101:b
97:a
89:a
85:b
80:a
76:b
71:b
67:a
20:)
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