Sequential analogues of the Lyapunov and Krein--Milman theorems in Fr\'echet spaces

F. S. Stonyakin

Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 57 (2015), pp. 162-183.

Résumé

In this paper we develop the theory of anti-compact sets we introduced earlier. We describe the class of Fréchet spaces where anti-compact sets exist. They are exactly the spaces that have a countable set of continuous linear functionals. In such spaces we prove an analogue of the Hahn–Banach theorem on extension of a continuous linear functional from the original space to a space generated by some anti-compact set. We obtain an analogue of the Lyapunov theorem on convexity and compactness of the range of vector measures, which establishes convexity and a special kind of relative weak compactness of the range of an atomless vector measure with values in a Fréchet space possessing an anti-compact set. Using this analogue of the Lyapunov theorem, we prove the solvability of an infinite-dimensional analogue of the problem of fair division of resources. We also obtain an analogue of the Lyapunov theorem for nonadditive analogues of measures that are vector quasi-measures valued in an infinite-dimensional Fréchet space possessing an anti-compact set. In the class of Fréchet spaces possessing an anti-compact set, we obtain analogues of the Krein–Milman theorem on extreme points for convex bounded sets that are not necessarily compact. A special place is occupied by analogues of the Krein–Milman theorem in terms of extreme sequences introduced in the paper (the so-called sequential analogues of the Krein–Milman theorem).

Bibliographie

[1] Arkin V. I., Levin V. L., “Vypuklost znachenii vektornykh integralov, teoremy izmerimogo vybora i variatsionnye zadachi”, Usp. mat. nauk, 27:3 (1972), 21–77 | MR | Zbl

[2] Balashov M. V., Polovinkin E. S., “M-silno vypuklye podmnozhestva i ikh porozhdayuschie podmnozhestva”, Mat. sb., 191:1 (2000), 27–64 | DOI | MR | Zbl

[3] Balashov M. V., “Ob analoge teoremy Kreina–Milmana dlya silno vypukloi obolochki v gilbertovom prostranstve”, Matematicheskie zametki, 71:1 (2002), 37–42 | DOI | MR | Zbl

[4] Vakhaniya N. N., Tarieladze V. I., Chobanyan S. A., Veroyatnostnye raspredeleniya v banakhovykh prostranstvakh, Nauka, M., 1985 | MR

[5] Ioffe A. D., Tikhomirov V. M., “Dvoistvennost vypuklykh funktsii i ekstremalnye zadachi”, Usp. mat. nauk, 23:6 (1968), 51–116 | MR | Zbl

[6] Kadets V. M., Kurs funktsionalnogo analiza, KhNU im. V. N. Karazina, Kharkov, 2006 | MR

[7] Kutateladze S. S., “Teorema Lyapunova, zonoidy i beng-beng”, Aleksei Andreevich Lyapunov. 100 let so dnya rozhdeniya, Akad. izd-vo “Geo”, Novosibirsk, 2011, 262–264

[8] Lyapunov A. A., “O vpolne additivnykh vektor-funktsiyakh”, Izv. AN SSSR. Ser. matem., 4:6 (1940), 465–478 | MR | Zbl

[9] Lyapunov A. A., “O vpolne additivnykh vektor-funktsiyakh. II”, Izv. AN SSSR. Ser. matem., 10:3 (1946), 277–279 | MR | Zbl

[10] Lyapunov A. N., “Teorema A. A. Lyapunova o vypuklosti znachenii mer”, Aleksei Andreevich Lyapunov. 100 let so dnya rozhdeniya, Akad. izd-vo “Geo”, Novosibirsk, 2011, 257–261

[11] Oben Zh.-P., Ekland I., Prikladnoi nelineinyi analiz, Mir, M., 1988 | MR

[12] Orlov I. V., “Gilbertovy kompakty, kompaktnye ellipsoidy i kompaktnye ekstremumy”, Sovrem. mat. Fundam. napravl., 29, 2008, 165–175 | MR

[13] Stonyakin F. S., “Silnye kompaktnye kharakteristiki i predelnaya forma svoistva Radona–Nikodima dlya vektornykh zaryadov so znacheniyami v prostranstvakh Freshe”, Uch. zap. Tavricheskogo natsionalnogo un-ta im. V. I. Vernadskogo. Ser. “Fiz.-mat. nauki”, 23(62):1 (2010), 131–149

[14] Stonyakin F. S., “Analog teoremy Ula o vypuklosti obraza vektornoi mery”, Dinam. sist., 3(31):3–4 (2013), 281–288

[15] Stonyakin F. S., “Antikompakty i ikh prilozheniya k analogam teorem Lyapunova i Lebega v prostranstvakh Freshe”, Sovrem. mat. Fundam. napravl., 53, 2014, 155–176

[16] Stonyakin F. S., “Sekventsialnaya versiya teoremy Ula o vypuklosti i kompaktnosti obraza vektornykh mer”, Uch. zap. Tavricheskogo natsionalnogo un-ta im. V. I. Vernadskogo. Ser. “Fiz.-mat. nauki”, 27(66):1 (2014), 100–111

[17] Stonyakin F. S., Magera M. V., “Rozv'yazannya zadachi pro rozdil skarbiv dlya dovilnoï kilkosti rozbiinikiv”, Uch. zap. Tavricheskogo natsionalnogo un-ta im. V. I. Vernadskogo. Ser. “Fiz.-mat. nauki”, 26(65):1 (2013), 109–128

[18] Stonyakin F. S., Shpilev R. O., “Analog teoremy Lyapunova o vypuklosti dlya $\varepsilon$-kvazimer i ee prilozheniya k zadache o razdele resursov”, Uch. zap. Tavricheskogo natsionalnogo un-ta im. V. I. Vernadskogo. Ser. “Fiz.-mat. nauki”, 27(66):1 (2014), 112–124

[19] Khille E., Fillips R., Funktsionalnyi analiz i polugruppy, IL, M., 1962

[20] Edvards E., Funktsionalnyi analiz. Teoriya i prilozheniya, Mir, M., 1969

[21] Arzi O., Aumann Y., Dombb Y., Throw one's cake – and eat it too, 2011, arXiv: 1101.4401v2[cs.GT]

[22] Chen Y., Lai J., Parkes D. C., Procaccia A. D., Truth, justice, and cake cutting, Association for the Advancement of Artificial Intelligence, 2010

[23] Dai P., Feinberg E. A., Extension of Lyapunov's convexity theorem to subranges, 2011, arXiv: 1102.2534v1[math.PR]

[24] Diestel J., Uhl J. J., Vector measures, Am. Math. Soc., Providence, 1977 | MR | Zbl

[25] Husseinov F., Sagarab N., “Concave measures and the fuzzy core of exchange economie with heterogeneous divisible commodities”, Fuzzy Sets and Systems, 198 (2012), 70–82 | DOI | MR | Zbl

[26] Maccheroni F., Marinacci M., “How to cut a pizza fairly: fair division with decreasing marginal evaluations”, Soc. Choice Welf., 20:3 (2003), 457–465 | DOI | MR | Zbl

[27] Mossel E., Tamuz O., Truthful fair division, 2010, arXiv: 1003.5480v2[cs.GT] | MR

[28] Neyman J., “Un théorème d'existence”, C. R. Math. Acad. Sci. Paris, 222 (1946), 843–845 | MR | Zbl

[29] Robertson J., Webb W., Cake-cutting algorithms: be fair if you can, AK Peters, Ltd., Natick, 1998 | MR | Zbl

[30] Steinhaus H., “Sur la division pragmatique”, Econometrica, 17 (1949), 315–319 | DOI | MR | Zbl