Existence et unicité des représentations intégrales dans les convexes compacts quelconques
Annales de l'Institut Fourier, Tome 13 (1963) no. 1, pp. 139-154.

Cet article est un exposé synthétique des résultats, obtenus ces dernières années, sur la représentation des points d’un convexe compact d’un espace vectoriel localement convexe séparé, comme barycentre d’une mesure positive portée, en un sens à préciser, par l’ensemble de ses points extrémaux.

On donne plusieurs caractérisations du cas où cette représentation est unique. Enfin on applique les résultats obtenus à l’étude de la frontière associée à un sous-espace vectoriel de C(Q) où Q est un espace compact.

@article{AIF_1963__13_1_139_0,
     author = {Choquet, Gustave and Meyer, Paul-Andr\'e},
     title = {Existence et unicit\'e des repr\'esentations int\'egrales dans les convexes compacts quelconques},
     journal = {Annales de l'Institut Fourier},
     pages = {139--154},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {13},
     number = {1},
     year = {1963},
     doi = {10.5802/aif.135},
     mrnumber = {26 #6748},
     zbl = {0122.34602},
     language = {fr},
     url = {https://www.numdam.org/articles/10.5802/aif.135/}
}
TY  - JOUR
AU  - Choquet, Gustave
AU  - Meyer, Paul-André
TI  - Existence et unicité des représentations intégrales dans les convexes compacts quelconques
JO  - Annales de l'Institut Fourier
PY  - 1963
SP  - 139
EP  - 154
VL  - 13
IS  - 1
PB  - Institut Fourier
PP  - Grenoble
UR  - https://www.numdam.org/articles/10.5802/aif.135/
DO  - 10.5802/aif.135
LA  - fr
ID  - AIF_1963__13_1_139_0
ER  - 
%0 Journal Article
%A Choquet, Gustave
%A Meyer, Paul-André
%T Existence et unicité des représentations intégrales dans les convexes compacts quelconques
%J Annales de l'Institut Fourier
%D 1963
%P 139-154
%V 13
%N 1
%I Institut Fourier
%C Grenoble
%U https://www.numdam.org/articles/10.5802/aif.135/
%R 10.5802/aif.135
%G fr
%F AIF_1963__13_1_139_0
Choquet, Gustave; Meyer, Paul-André. Existence et unicité des représentations intégrales dans les convexes compacts quelconques. Annales de l'Institut Fourier, Tome 13 (1963) no. 1, pp. 139-154. doi : 10.5802/aif.135. https://www.numdam.org/articles/10.5802/aif.135/

[1] H. Bauer, Minimalstellen von Funktionen und Extremal-punkte II, Arch. Math. 11, 1960, pp. 200-205. | MR | Zbl

[2] H. Bauer. Silovscher Rand und Dirichletsches Problem, Ann. Inst. Fourrier, Grenoble, t. 11, 1961, pp. 89-136. | Numdam | MR | Zbl

[3] E. Bishop et K. De Leeuw, The representation of linear functionals by measures on sets of extreme points, Ann. Inst. Fourier, Grenoble, t. 9, 1959, pp. 305-331. | Numdam | MR | Zbl

[4] F. F. Bonsall, On the representation of the points of a convex set, J. London math. Soc. (à paraître). | Zbl

[5] G. Choquet, Existence et unicité des représentations intégrales, Séminaire Bourbaki, Décembre 1956, 139, 15 pages. | Numdam | Zbl

[6] G. Choquet, Le Théorème de représentation intégrale dans les ensembles convexes compacts, Ann. Inst. Fourier, Grenoble, t. 10, 1960, pp. 333-344. | Numdam | MR | Zbl

[7] G. Choquet. Forme abstraite du théorème de capacitabilité, Ann. Inst. Fourier, Grenoble, t. 9, 1959, pp. 83-89. | Numdam | MR | Zbl

[8] G. Choquet, Remarques à propos de la démonstration d'unicité de P. A. Meyer, Séminaire du Potentiel, t. 6, 1962, n°8, 13 pages. | Numdam | Zbl

[9] M. Hervé, Sur les représentations intégrales à l'aide des points extrémaux, C. R. Acad. Sc., Paris, t. 253, 1961, p. 366-368. | MR | Zbl

[10] P. A. Meyer, Sur les démonstrations nouvelles du théorème de Choquet, Séminaire du Potentiel, t. 6, 1962, n°7, 9 pages. | Numdam | MR | Zbl

[11] G. Mokobodzki, Balayage défini par un cône convexe de fonctions, C. R. Acad. Sc., Paris, t. 254, 1962, p. 803-805. | MR | Zbl

[19] G. Mokobodzki, Principe du balayage, principe de domination, Séminaire d'initiation à l'Analyse, 1re année, 1962, n°1, 11 pages. | Numdam

  • Manivannan, Varadha Raj; Venkataraman, Madhu Helmholtz operators on infinite graphs, Heliyon, Volume 10 (2024) no. 2, p. e24761 | DOI:10.1016/j.heliyon.2024.e24761
  • Bachir, Mohammed; Blot, Joël Lagrange Multipliers in Locally Convex Spaces, Journal of Optimization Theory and Applications, Volume 201 (2024) no. 3, p. 1275 | DOI:10.1007/s10957-024-02428-z
  • Fidaleo, Francesco Symmetric states for C∗-Fermi systems, Reviews in Mathematical Physics, Volume 34 (2022) no. 09 | DOI:10.1142/s0129055x22500301
  • Dor-On, Adam; Eilers, Søren; Geffen, Shirly Classification of irreversible and reversible Pimsner operator algebras, Compositio Mathematica, Volume 156 (2020) no. 12, p. 2510 | DOI:10.1112/s0010437x2000754x
  • Shen, Jie; Shen, Yi; Wang, Bin; Wang, Ruodu Distributional compatibility for change of measures, Finance and Stochastics, Volume 23 (2019) no. 3, p. 761 | DOI:10.1007/s00780-019-00393-4
  • Guerdjikova, Ani; Quiggin, John Supplement to 'Market Selection with Differential Financial Constraints', SSRN Electronic Journal (2019) | DOI:10.2139/ssrn.3372292
  • Sibony, Nessim Pfaff systems, currents and hulls, Mathematische Zeitschrift, Volume 285 (2017) no. 3-4, p. 1107 | DOI:10.1007/s00209-016-1740-8
  • Ernst, Emil; Zaffaroni, Alberto Characterizing Sets of Lower Bounds: a Hidden Convexity Result, Set-Valued and Variational Analysis, Volume 25 (2017) no. 4, p. 639 | DOI:10.1007/s11228-017-0416-9
  • Cassese, Gianluca Conglomerability and Representations, SSRN Electronic Journal (2015) | DOI:10.2139/ssrn.2704508
  • Hauray, Maxime; Mischler, Stéphane On Kac's chaos and related problems, Journal of Functional Analysis, Volume 266 (2014) no. 10, p. 6055 | DOI:10.1016/j.jfa.2014.02.030
  • Henry-Labordere, Pierre Linking Vanillas and VIX Options: A Constrained Martingale Optimal Transport, SSRN Electronic Journal (2013) | DOI:10.2139/ssrn.2354898
  • Levin, Leonid A., 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science (2012), p. 510 | DOI:10.1109/focs.2012.50
  • Postolică, Vasile Choquet boundaries and efficiency, Computers Mathematics with Applications, Volume 55 (2008) no. 3, p. 381 | DOI:10.1016/j.camwa.2007.03.022
  • Postolică, Vasile Efficiency and Applications, PAMM, Volume 8 (2008) no. 1, p. 10783 | DOI:10.1002/pamm.200810783
  • Ruelle, David Thermodynamic Formalism, 2004 | DOI:10.1017/cbo9780511617546
  • Joe, Diestel; Johan, swart The Riesz Theorem, Handbook of Measure Theory (2002), p. 401 | DOI:10.1016/b978-044450263-6/50010-5
  • Fonf, V.P.; Lindenstrauss, J.; Phelps, R.R. Infinite Dimensional Convexity, Volume 1 (2001), p. 599 | DOI:10.1016/s1874-5849(01)80017-6
  • Kružík, Martin; Roubíček, Tomáš Some geometric properties of the set of generalized Young functionals, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Volume 129 (1999) no. 3, p. 601 | DOI:10.1017/s0308210500021521
  • Bibliography, Korovkin-type Approximation Theory and its Applications (1994), p. 555 | DOI:10.1515/9783110884586.555
  • Klein, David; Yang, Wei-Shih Absence of first-order phase transitions for antiferromagnetic systems, Journal of Statistical Physics, Volume 70 (1993) no. 5-6, p. 1391 | DOI:10.1007/bf01049441
  • Bibliography, Theory of Convex Structures, Volume 50 (1993), p. 507 | DOI:10.1016/s0924-6509(09)70386-1
  • Arkeryd, L.; Caprino, S.; Ianiro, N. The homogeneous Boltzmann hierarchy and statistical solutions to the homogeneous Boltzmann equation, Journal of Statistical Physics, Volume 63 (1991) no. 1-2, p. 345 | DOI:10.1007/bf01026609
  • References, Elements of Differentiable Dynamics and Bifurcation Theory (1989), p. 172 | DOI:10.1016/b978-0-12-601710-6.50007-9
  • Wagenknecht, Niels Die zweite randwertaufgabe der potentialtheorie als duales gegenstück zur ersten, Complex Variables, Theory and Application: An International Journal, Volume 10 (1988) no. 4, p. 249 | DOI:10.1080/17476938808814304
  • Bibliography, Probabilities and Potential C - Potential Theory for Discrete and Continuous Semigroups, Volume 151 (1988), p. 387 | DOI:10.1016/s0304-0208(08)71921-2
  • Teleman, Silviu A lattice-theoretic characterization of choquet simplexes, Operator Algebras and their Connections with Topology and Ergodic Theory, Volume 1132 (1985), p. 517 | DOI:10.1007/bfb0074907
  • Batty, Charles J. K.; Robinson, Derek W. Positive one-parameter semigroups on ordered banach spaces, Acta Applicandae Mathematicae, Volume 2 (1984) no. 3-4, p. 221 | DOI:10.1007/bf02280855
  • Batty, Charles J. K.; Robinson, Derek W. Positive One-Parameter Semigroups on Ordered Banach Spaces, Positive Semigroups of Operators, and Applications (1984), p. 221 | DOI:10.1007/978-94-009-6484-6_2
  • Ríos Insua, M. Jesús “Sobre los modelos jerárquicos y su relación con el problema de decisión con información parcial a priori”, Trabajos de Estadistica y de Investigacion Operativa, Volume 35 (1984) no. 2, p. 222 | DOI:10.1007/bf02888903
  • References, Convex Cones, Volume 56 (1981), p. 403 | DOI:10.1016/s0304-0208(08)71030-2
  • References, Convexity Theory and its Applications in Functional Analysis (1980), p. 252 | DOI:10.1016/b978-0-12-065340-9.50011-5
  • Weizsäcker, H.V.; Winkler, G. Non-Compact Extremal Integral Representations: Some Probabilistic Aspects, Functional Analysis: Surveys and Recent Results II, Volume 38 (1980), p. 115 | DOI:10.1016/s0304-0208(08)70805-3
  • Olsen, Gunnar Hans On Simplices and the Poulsen Simplex, Functional Analysis: Surveys and Recent Results II, Volume 38 (1980), p. 31 | DOI:10.1016/s0304-0208(08)70799-0
  • Bratteli, Ola; Robinson, Derek W. Decomposition Theory, Operator Algebras and Quantum Statistical Mechanics (1979), p. 309 | DOI:10.1007/978-3-662-02313-6_4
  • SELECTED BIBLIOGRAPHY, Measure and Integral (1978), p. 549 | DOI:10.1016/b978-0-12-378550-3.50023-2
  • Ruelle, David Integral representation of measures associated with a foliation, Publications mathématiques de l'IHÉS, Volume 48 (1978) no. 1, p. 127 | DOI:10.1007/bf02684314
  • Cassandro, M.; Olivieri, E.; Pellegrinotti, A.; Presutti, E. Existence and uniqueness of DLR measures for unbounded spin systems, Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, Volume 41 (1978) no. 4, p. 313 | DOI:10.1007/bf00533602
  • Castaing, Charles; Valadier, Michel Topological property of the profile of a measurable multifunction with compact convex values, Convex Analysis and Measurable Multifunctions, Volume 580 (1977), p. 91 | DOI:10.1007/bfb0087689
  • Ruelle, David Formalisme thermodynamique, Séminaire Bourbaki vol. 1975/76 Exposés 471–488, Volume 567 (1977), p. 154 | DOI:10.1007/bfb0096067
  • Watanabe, Hisako Simplexes and Dirichlet problems on locally compact spaces, Hiroshima Mathematical Journal, Volume 6 (1976) no. 2 | DOI:10.32917/hmj/1206136333
  • Landsberg, M.; Schirotzek, W. Extremal and Invariant Extensions of Linear Functionals, Mathematische Nachrichten, Volume 71 (1976) no. 1, p. 191 | DOI:10.1002/mana.19760710115
  • Shashkin, Yu. A. Convex sets, extreme points, and simplexes, Journal of Soviet Mathematics, Volume 4 (1975) no. 6, p. 625 | DOI:10.1007/bf01083882
  • Goullet de Rugy, Alain Simplex spaces, Foundations of Quantum Mechanics and Ordered Linear Spaces, Volume 29 (1974), p. 33 | DOI:10.1007/3-540-06725-6_6
  • Choquet, Gustave; Deny, Jacques Noyaux de convolution et balayage sur tout ouvert, Théorie du Potentiel et Analyse Harmonique, Volume 404 (1974), p. 60 | DOI:10.1007/bfb0060610
  • Flato, Moshé A Short Survey of Modern Statistical Mechanics, Studies in Mathematical Physics (1973), p. 113 | DOI:10.1007/978-94-010-2669-7_4
  • Fakhoury, Hicham Préduaux de L-espace: Notion de Centre, Journal of Functional Analysis, Volume 9 (1972) no. 2, p. 189 | DOI:10.1016/0022-1236(72)90010-9
  • Ruelle, D. Superstable interactions in classical statistical mechanics, Communications in Mathematical Physics, Volume 18 (1970) no. 2, p. 127 | DOI:10.1007/bf01646091
  • Ruelle, David Integral representation of states on a C∗-algebra, Journal of Functional Analysis, Volume 6 (1970) no. 1, p. 116 | DOI:10.1016/0022-1236(70)90051-0
  • Krause, Ulrich Der Satz von Choquet als ein abstrakter Spektralsatz und vice versa, Mathematische Annalen, Volume 184 (1970) no. 4, p. 275 | DOI:10.1007/bf01350856
  • Maltese, George On Bauer's characterization of extreme points, Mathematische Annalen, Volume 184 (1970) no. 4, p. 326 | DOI:10.1007/bf01350861
  • Lanford, O. E. The classical mechanics of one-dimensional systems of infinitely many particles, Communications in Mathematical Physics, Volume 11 (1969) no. 4, p. 257 | DOI:10.1007/bf01645848
  • Lanford, O. E.; Ruelle, D. Observables at infinity and states with short range correlations in statistical mechanics, Communications in Mathematical Physics, Volume 13 (1969) no. 3, p. 194 | DOI:10.1007/bf01645487
  • Doplicher, S; Kastler, D; Størmer, E Invariant states and asymptotic abelianness, Journal of Functional Analysis, Volume 3 (1969) no. 3, p. 419 | DOI:10.1016/0022-1236(69)90033-0
  • Alfsen, Erik M. On the Dirichlet problem of the Choquet boundary, Acta Mathematica, Volume 120 (1968) no. 0, p. 149 | DOI:10.1007/bf02394608
  • Bichteler, Klaus On the existence of noncontinuous irreducible representations of a locally compact group, Inventiones Mathematicae, Volume 6 (1968) no. 2, p. 159 | DOI:10.1007/bf01425453
  • Rogalski, Marc Opérateurs de Lion, projecteurs boréliens et simplexes analytiques, Journal of Functional Analysis, Volume 2 (1968) no. 4, p. 458 | DOI:10.1016/0022-1236(68)90005-0
  • Lanford, Oscar E.; Robinson, Derek W. Mean Entropy of States in Quantum-Statistical Mechanics, Journal of Mathematical Physics, Volume 9 (1968) no. 7, p. 1120 | DOI:10.1063/1.1664685
  • Jellett, Francis Homomorphisms and inverse limits of Choquet simplexes, Mathematische Zeitschrift, Volume 103 (1968) no. 3, p. 219 | DOI:10.1007/bf01111040
  • Grossman, Marvin W. An extremal property for certain spaces of lower semi-continuous functions, Mathematische Zeitschrift, Volume 106 (1968) no. 2, p. 139 | DOI:10.1007/bf01110720
  • Davies, E. B. The structure and ideal theory of the predual of a Banach lattice, Transactions of the American Mathematical Society, Volume 131 (1968) no. 2, p. 544 | DOI:10.1090/s0002-9947-1968-0222604-8
  • Lazar, Aldo J. Spaces of affine continuous functions on simplexes, Transactions of the American Mathematical Society, Volume 134 (1968) no. 3, p. 503 | DOI:10.1090/s0002-9947-1968-0233188-2
  • Effros, Edward G. Structure in simplexes, Acta Mathematica, Volume 117 (1967) no. 0, p. 103 | DOI:10.1007/bf02395042
  • Størmer, Erling Large groups of automorphisms of C*-algebras, Communications in Mathematical Physics, Volume 5 (1967) no. 1, p. 1 | DOI:10.1007/bf01646355
  • Robinson, Derek W.; Ruelle, David Mean entropy of states in classical statistical mechanics, Communications in Mathematical Physics, Volume 5 (1967) no. 4, p. 288 | DOI:10.1007/bf01646480
  • Thoma, Elmar Zur harmonischen Analyse klassenfiniter Gruppen, Inventiones Mathematicae, Volume 3 (1967) no. 1, p. 20 | DOI:10.1007/bf01425489
  • Lazar, A. J. Affine functions on simplexes and extreme operators, Israel Journal of Mathematics, Volume 5 (1967) no. 1, p. 31 | DOI:10.1007/bf02771594
  • Effros, Edward G Structure in simplexes. II, Journal of Functional Analysis, Volume 1 (1967) no. 4, p. 379 | DOI:10.1016/0022-1236(67)90008-0
  • Khanh, Bui Doan Sur la Φ-convexité de Ky Fan, Journal of Mathematical Analysis and Applications, Volume 20 (1967) no. 1, p. 188 | DOI:10.1016/0022-247x(67)90114-x
  • Lanford, O.; Ruelle, D. Integral Representations of Invariant States on B* Algebras, Journal of Mathematical Physics, Volume 8 (1967) no. 7, p. 1460 | DOI:10.1063/1.1705360
  • Ruelle, D. States of Classical Statistical Mechanics, Journal of Mathematical Physics, Volume 8 (1967) no. 8, p. 1657 | DOI:10.1063/1.1705406
  • Ruelle, David States of physical systems, Communications in Mathematical Physics, Volume 3 (1966) no. 2, p. 133 | DOI:10.1007/bf01645450
  • Thoma, Elmar Invariante positiv definite Klassenfunktionen und ergodische Maße, Contributions to Functional Analysis (1966), p. 172 | DOI:10.1007/978-3-642-85997-7_8
  • Bucy, R. S.; Maltese, G. A Representation Theorem for Positive Functionals on Involution Algebras, Contributions to Functional Analysis (1966), p. 364 | DOI:10.1007/978-3-642-85997-7_24
  • Bucy, R. S.; Maltese, G. A representation theorem for positive functionals on involution algebras, Mathematische Annalen, Volume 162 (1966) no. 3, p. 364 | DOI:10.1007/bf01369109
  • Brown, C. C. Zur Verallgemeinerung der abstrakten Integrationstheorie, Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, Volume 6 (1966) no. 4, p. 317 | DOI:10.1007/bf00537830
  • Thoma, Elmar Invariante positiv definite Klassenfunktionen und ergodische Ma�e, Mathematische Annalen, Volume 162 (1965) no. 1, p. 172 | DOI:10.1007/bf01361942
  • Eggleston, H. G.; Grünbaum, Branko; Klee, Victor Some semicontinuity theorems for convex polytopes and cell-complexes, Commentarii Mathematici Helvetici, Volume 39 (1964) no. 1, p. 165 | DOI:10.1007/bf02566949
  • Bauer, Heinz Kennzeichnung kompakter Simplexe mit abgeschlossener Extremalpunktmenge, Archiv der Mathematik, Volume 14 (1963) no. 1, p. 415 | DOI:10.1007/bf01234978

Cité par 78 documents. Sources : Crossref