Eidetic Reduction of Information Geometry Through Legendre Duality of Koszul Characteristic Function and Entropy: From Massieu–Duhem Potentials to Geometric Souriau Temperature and Balian Quantum Fisher Metric

  • Frédéric BarbarescoEmail author
Part of the Signals and Communication Technology book series (SCT)


Based on Koszul theory of sharp convex cone and its hessian geometry, Information Geometry metric is introduced by Koszul form as hessian of Koszul–Vinberg Characteristic function logarithm (KVCFL). The front of the Legendre mapping of this KVCFL is the graph of a convex function, the Legendre transform of this KVCFL. By analogy in thermodynamic with Dual Massieu–Duhem potentials (Free Energy and Entropy), the Legendre transform of KVCFL is interpreted as a “Koszul Entropy”. This Legendre duality is considered in more general framework of Contact Geometry, the odd-dimensional twin of symplectic geometry, with Legendre fibration and mapping. Other analogies will be introduced with large deviation theory with Cumulant Generating and Rate functions (Legendre duality by Laplace Principle) and with Legendre duality in Mechanics between Hamiltonian and Lagrangian. In all these domains, we observe that the “Characteristic function” and its derivatives capture all information of random variable, system or physical model. We present two extensions of this theory with Souriau’s Geometric Temperature deduced from covariant definition of thermodynamic equilibriums, and with Balian quantum Fisher metric defined and built as hessian of von Neumann entropy. Finally, we apply Koszul geometry for Symmetric/Hermitian Positive Definite Matrices cones, and more particularly for covariance matrices of stationary signal that are characterized by specific matrix structures: Toeplitz Hermitian Positive Definite Matrix structure (covariance matrix of a stationary time series) or Toeplitz-Block-Toeplitz Hermitian Positive Definite Matrix structure (covariance matrix of a stationary space–time series). By extension, we introduce a new geometry for non-stationary signal through Fréchet metric space of geodesic paths on structured matrix manifolds. We conclude with extensions towards two general concepts of “Generating Inner Product” and “Generating Function”.


Koszul characteristic function Koszul entropy Koszul forms Laplace principle Massieu–Duhem potential Projective legendre duality Contact geometry Information geometry Souriau geometric temperature Balian quantum information metric Cartan–Siegel homogeneous bounded domains Generating function 


  1. 1.
    Koszul, J.L.: Variétés localement plates et convexité. Osaka J. Math. 2, 285–290 (1965)Google Scholar
  2. 2.
    Vey, J.: Sur les automorphismes affines des ouverts convexes saillants. Annali della Scuola Normale Superiore di Pisa, Classe di Science, 3e série, Tome 24(4), 641–665 (1970)Google Scholar
  3. 3.
    Massieu, F.: Sur les fonctions caractéristiques des divers fluides. C. R. Acad. Sci. 69, 858–862 (1869)Google Scholar
  4. 4.
    Massieu, F.: Addition au précédent Mémoire sur les fonctions caractéristiques. C. R. Acad. Sci. 69, 1057–1061 (1869)Google Scholar
  5. 5.
    Massieu, F.: Thermodynamique: mémoire sur les fonctions caractéristiques des divers fluides et sur la théorie des vapeurs, 92 p. Académie des Sciences (1876)Google Scholar
  6. 6.
    Duhem, P.: Sur les équations générales de la thermodynamique. Annales Scientifiques de l’Ecole Normale Supérieure, 3e série, Tome 8, 231 (1891)Google Scholar
  7. 7.
    Duhem, P.: Commentaire aux principes de la thermodynamique. Première partie, Journal de Mathématiques pures et appliquées, 4e série, Tome 8, 269 (1892)Google Scholar
  8. 8.
    Duhem, P.: Commentaire aux principes de la thermodynamique—troisième partie. Journal de Mathématiques pures et appliquées, 4e série, Tome 10, 203 (1894)Google Scholar
  9. 9.
    Duhem, P.: Les théories de la chaleur. Duhem 1992, 351–1 (1895)Google Scholar
  10. 10.
    Laplace, P.S.: Mémoire sur la probabilité des causes sur les évènements. Mémoires de Mathématique et de Physique, Tome Sixième (1774)Google Scholar
  11. 11.
    Arnold, V.I., Givental, A.G.: Symplectic geometry. In: Encyclopedia of Mathematical Science, vol. 4. Springer, New York (translated from Russian) (2001)Google Scholar
  12. 12.
    Fitzpatrick, S.: On the geometric quantization of contact manifolds. (2013). Accessed Feb 2013
  13. 13.
    Rajeev, S.G.: Quantization of contact manifolds and thermodynamics. Ann. Phys. 323(3), 768–82 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Gibbs, J.W.: Graphical methods in the thermodynamics of fluids. In: Bumstead, H.A., Van Name, R.G. (eds.) Scientific Papers of J Willard Gibbs, 2 vols. Dover, New York (1961)Google Scholar
  15. 15.
    Dedecker, P.: A property of differential forms in the calculus of variations. Pac. J. Math. 7(4), 1545–9 (1957)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Lepage, T.: Sur les champs géodésiques du calcul des variations. Bull. Acad. Roy. Belg. Cl. Sci. 27, 716–729, 1036–1046 (1936)Google Scholar
  17. 17.
    Mrugala, R.: On contact and metric structures on thermodynamic spaces. RIMS Kokyuroku 1142, 167–81 (2000)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Ingarden R.S., Kossakowski A.: The poisson probability distribution and information thermodynamics. Bull. Acad. Pol. Sci. Sér. Sci. Math. Astron. Phys. 19, 83–85 (1971)Google Scholar
  19. 19.
    Ingarden, R.S.: Information geometry in functional spaces of classical and quantum finite statistical systems. Int. J. Eng. Sci. 19(12), 1609–33 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Ingarden, R.S., Janyszek, H.: On the local Riemannian structure of the state space of classical information thermodynamics. Tensor, New Ser. 39, 279–85 (1982)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Ingarden, R.S., Kawaguchi, M., Sato, Y.: Information geometry of classical thermodynamical systems. Tensor, New Ser. 39, 267–78 (1982)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Ingarden R.S.: Information geometry of thermodynamics. In: Transactions of the Tenth Prague Conference Czechoslovak Academy of Sciences, vol. 10A–B, pp. 421–428 (1987)Google Scholar
  23. 23.
    Ingarden, R.S.: Information geometry of thermodynamics, information theory, statistical decision functions, random processes. In: Transactions of the 10th Prague Conference, Prague/Czechoslovakia 1986, vol. A, pp. 421–428 (1988)Google Scholar
  24. 24.
    Ingarden, R.S., Nakagomi, T.: The second order extension of the Gibbs state. Open Syst. Inf. Dyn. 1(2), 243–58 (1992)CrossRefzbMATHGoogle Scholar
  25. 25.
    Arnold V.I.: Contact geometry: the geometrical method of Gibbs’s thermodynamics. In: Proceedings of the Gibbs Symposium, pp. 163–179. American Mathematical Society, Providence, RI (1990)Google Scholar
  26. 26.
    Cartan, E.: Leçons sur les Invariants Intégraux. Hermann, Paris (1922)zbMATHGoogle Scholar
  27. 27.
    Koszul J.L.: Exposés sur les espaces homogènes symétriques. Publicação da Sociedade de Matematica de São Paulo (1959)Google Scholar
  28. 28.
    Koszul J.L.: Sur la forme hermitienne canonique des espaces homogènes complexes. Can. J. Math. 7(4), 562–576 (1955)Google Scholar
  29. 29.
    Koszul, J.L.: Lectures on Groups of Transformations. Tata Institute of Fundamental Research, Bombay (1965)zbMATHGoogle Scholar
  30. 30.
    Koszul, J.L.: Domaines bornées homogènes et orbites de groupes de transformations affines. Bull. Soc. Math. Fr. 89, 515–33 (1961)zbMATHMathSciNetGoogle Scholar
  31. 31.
    Koszul, J.L.: Ouverts convexes homogènes des espaces affines. Math. Z. 79, 254–9 (1962)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Koszul, J.L.: Déformations des variétés localement plates. Ann. Inst. Fourier 18, 103–14 (1968)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Vinberg, E.: Homogeneous convex cones. Trans. Moscow Math. Soc. 12, 340–363 (1963)Google Scholar
  34. 34.
    Vesentini E.: Geometry of Homogeneous Bounded Domains. Springer, Berlin (2011). Reprint of the 1st Edn. C.I.M.E., Ed. Cremonese, Roma (1968)Google Scholar
  35. 35.
    Barbaresco F.: Information geometry of covariance matrix: Cartan-Siegel homogeneous bounded domains, Mostow/Berger fibration and Fréchet median. In: Bhatia, R., Nielsen, F. (eds.) Matrix Information Geometry, pp. 199–256. Springer, New York (2012)Google Scholar
  36. 36.
    Arnaudon M., Barbaresco F., Le, Y.: Riemannian medians and means with applications to radar signal processing. IEEE J. Sel. Top. Sig. Process. 7(4), 595–604 (2013)Google Scholar
  37. 37.
    Dorfmeister, J.: Inductive construction of homogeneous cones. Trans. Am. Math. Soc. 252, 321–49 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Dorfmeister, J.: Homogeneous siegel domains. Nagoya Math. J. 86, 39–83 (1982)zbMATHMathSciNetGoogle Scholar
  39. 39.
    Poincaré, H.: Thermodynamique, Cours de Physique Mathématique. G. Carré, Paris (1892)Google Scholar
  40. 40.
    Poincaré, H.: Calcul des Probabilités. Gauthier-Villars, Paris (1896)zbMATHGoogle Scholar
  41. 41.
    Faraut, J., Koranyi, A.: Analysis on Symmetric Cones. The Clarendon Press, New York (1994)zbMATHGoogle Scholar
  42. 42.
    Faraut, J., Koranyi, A.: Oxford Mathematical Monographs. Oxford University Press, New York (1994)Google Scholar
  43. 43.
    Varadhan, S.R.S.: Asymptotic probability and differential equations. Commun. Pure Appl. Math. 19, 261–86 (1966)CrossRefzbMATHMathSciNetGoogle Scholar
  44. 44.
    Sanov, I.N.: On the probability of large deviations of random magnitudes. Mat. Sb. 42(84), 11–44 (1957)MathSciNetGoogle Scholar
  45. 45.
    Ellis, R.S.: The Theory of Large Deviations and Applications to Statistical Mechanics. Lecture Notes for Ecole de Physique Les Houches, France (2009)Google Scholar
  46. 46.
    Touchette, H.: The large deviation approach to statistical mechanics. Phys. Rep. 478(1—-3), 1–69 (2009)CrossRefMathSciNetGoogle Scholar
  47. 47.
    Cartan, E.: Sur les domaines bornés de l’espace de n variables complexes. Abh. Math. Semin. Hamburg 1, 116–62 (1935)CrossRefGoogle Scholar
  48. 48.
    Lichnerowicz, A.: Espaces homogènes Kähleriens. In: Collection Géométrie Différentielle, pp. 171–84, Strasbourg (1953)Google Scholar
  49. 49.
    Sasaki T.: A note on characteristic functions and projectively invariant metrics on a bounded convex domain. Tokyo J. Math. 8(1), 49–79 (1985)Google Scholar
  50. 50.
    Sasaki, T.: Hyperbolic affine hyperspheres. Nagoya Math. J. 77, 107–23 (1980)zbMATHMathSciNetGoogle Scholar
  51. 51.
    Trench W.F.: An algorithm for the inversion of finite Toeplitz matrices. J. Soc. Ind. Appl. Math. 12, 515–522 (1964)Google Scholar
  52. 52.
    Verblunsky, S.: On positive harmonic functions. Proc. London Math. Soc. 38, 125–57 (1935)CrossRefMathSciNetGoogle Scholar
  53. 53.
    Verblunsky, S.: On positive harmonic functions. Proc. London Math. Soc. 40, 290–20 (1936)CrossRefMathSciNetGoogle Scholar
  54. 54.
    Hauser, R.A., Güler, O.: Self-scaled barrier functions on symmetric cones and their classification. Found. Comput. Math. 2(2), 121–43 (2002)zbMATHMathSciNetGoogle Scholar
  55. 55.
    Vinberg, E.B.: Structure of the group of automorphisms of a homogeneous convex cone. Tr. Mosk. Mat. O-va 13, 56–83 (1965)zbMATHMathSciNetGoogle Scholar
  56. 56.
    Siegel, C.L.: Über der analytische theorie der quadratischen Formen. Ann. Math. 36, 527–606 (1935)CrossRefGoogle Scholar
  57. 57.
    Duan, X., Sun, H., Peng, L.: Riemannian means on special euclidean group and unipotent matrices group. Sci. World J. 2013, ID 292787 (2013)Google Scholar
  58. 58.
    Soize C.: A nonparametric model of random uncertainties for reduced matrix models in structural dynamics. Probab. Eng. Mech. 15(3), 277–294 (2000)Google Scholar
  59. 59.
    Bennequin, D.: Dualités de champs et de cordes. Séminaire N. Bourbaki, exp. no. 899, pp. 117–148 (2001–2002)Google Scholar
  60. 60.
    Bennequin D.: Dualité Physique-Géométrie et Arithmétique, Brasilia (2012)Google Scholar
  61. 61.
    Chasles M.: Aperçu historique sur l’origine et le développement des méthodes en géométrie (1837)Google Scholar
  62. 62.
    Gergonne, J.D.: Polémique mathématique. Réclamation de M. le capitaine Poncelet (extraite du bulletin universel des annonces et nouvelles scientifiques); avec des notes. Annales de Gergonne, vol. 18, pp. 125–125. (1827–1828)
  63. 63.
    Poncelet, J.V.: Traité des propriétés projectives des figures (1822)Google Scholar
  64. 64.
    André, Y.: Dualités. Sixième séance, ENS, Mai (2008)Google Scholar
  65. 65.
    Atiyah, M.F.: Duality in mathematics and physics, lecture Riemann’s influence in geometry. Analysis and Number Theory at the Facultat de Matematiques i Estadıstica of the Universitat Politecnica de Catalunya (2007)Google Scholar
  66. 66.
    Von Oettingen, A.J.: Harmoniesystem in dualer Entwicklung. Studien zur Theorie der Musik, Dorpat und Leipzig (1866)Google Scholar
  67. 67.
    Von Oettingen, A.J.: Das duale system der harmonie. In: Annalen der Naturphilosophie, vol. 1 (1902)Google Scholar
  68. 68.
    Von Oettingen, A.J.: Das duale system der harmonie. In: Annalen der Naturphilosophie, vol. 2, pp. 62–75 (1903/1904)Google Scholar
  69. 69.
    Von Oettingen, A.J.: Das duale system der harmonie. In: Annalen der Naturphilosophie, vol. 3, pp. 375–403 (1904)Google Scholar
  70. 70.
    Von Oettingen, A.J.: Das duale system der harmonie. In: Annalen der Naturphilosophie, vol. 4, pp. 241–269 (1905)Google Scholar
  71. 71.
    Von Oettingen, A.J.: Das duale system der harmonie. In: Annalen der Naturphilosophie, vol. 5, pp. 116–152, 301–338, 449–503 (1906)Google Scholar
  72. 72.
    Von Oettingen, A.J.: Das duale Harmoniesystem, Leipzig (1913)Google Scholar
  73. 73.
    Von Oettingen, A.J.: Die Grundlagen der musikwissenschaft und das duale reinistrument. In: Abhand-lungen der mathematisch-physikalischen Klasse der Königlich Sächsischen Gesell-schaft der Wissenschaften, vol. 34, pp. S.I–XVI, 155–361 (1917)Google Scholar
  74. 74.
    D’Alembert, J.R.: Éléments de musique, théorique et pratique, suivant les principes de M. Rameau, Paris (1752)Google Scholar
  75. 75.
    Rameau, J.P.: Traité de l’harmonie Réduite à ses Principes Naturels. Ballard, Paris (1722)Google Scholar
  76. 76.
    Rameau, J.P.: Nouveau système de Musique Théorique. Ballard, Paris (1726)Google Scholar
  77. 77.
    Yang, L.: Médianes de mesures de probabilité dans les variétés riemanniennes et applications à la détection de cibles radar. Thèse de l’Université de Poitiers, tel-00664188, 2011, Thales PhD Award (2012)Google Scholar
  78. 78.
    Barbaresco, F.: Algorithme de Burg Régularisé FSDS. Comparaison avec l’algorithme de Burg MFE, pp. 29–32 GRETSI conference (1995)Google Scholar
  79. 79.
    Barbaresco, F.: Information geometry of covariance matrix. In: Nielsen, F., Bhatia, R. (eds.) Matrix Information Geometry Book. Springer, Berlin (2012)Google Scholar
  80. 80.
    Émery, M., Mokobodzki, G.: Sur le barycentre d’une probabilité dans une variété. Séminaire de probabilité Strasbourg 25, 220–233 (1991)Google Scholar
  81. 81.
    Friedrich, T.: Die Fisher-information und symplektische strukturen. Math. Nachr. 153, 273–96 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  82. 82.
    Bingham N.H.: Szegö’s Theorem and Its Probabilistic Descendants. (2012)
  83. 83.
    Landau, H.J.: Maximum entropy and the moment problem. Bull. Am. Math. Soc. 16(1), 47–77 (1987)Google Scholar
  84. 84.
    Siegel, C.L.: Symplectic geometry. Am. J. Math. 65, 1–86 (1943)CrossRefGoogle Scholar
  85. 85.
    Libermann, P., Marle, C.M.: Symplectic Geometry and Analytical Mechanics. Reidel, Dordrecht (1987)CrossRefzbMATHGoogle Scholar
  86. 86.
    Delsarte, P., Genin, Y.V.: Orthogonal polynomial matrices on the unit circle. IEEE Trans. Comput. Soc. 25(3), 149–160 (1978)Google Scholar
  87. 87.
    Kanhouche, R.: A modified burg algorithm equivalent. In: Results to Levinson algorithm.
  88. 88.
    Douady, C.J., Earle, C.J.: Conformally natural extension of homeomorphisms of circle. Acta Math. 157, 23–48 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  89. 89.
    Balian, R.: A metric for quantum states issued from von Neumann’s entropy. In: Nielsen, F., Barbaresco, F. (Eds.) Geometric Science of Information. Lecture Notes in Computer Science, vol. 8085, pp. 513–518Google Scholar
  90. 90.
    Balian, R.: Incomplete descriptions and relevant entropies. Am. J. Phys. 67, 1078–90 (1989)CrossRefMathSciNetGoogle Scholar
  91. 91.
    Balian, R.: Information in statistical physics. Stud. Hist. Philos. Mod. Phys. 36, 323–353 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  92. 92.
    Allahverdyan, A., Balian, R., Nieuwenhuizen, T.: Understanding quantum measurement from the solution of dynamical models. Phys. Rep. 525, 1–166 (2013) (ArXiv: 1107, 2138)Google Scholar
  93. 93.
    Balian, R.: From Microphysics to Macrophysics: Methods and Applications of Statistical Physics, vol. 1–2. Springer (2007)Google Scholar
  94. 94.
    Balian, R., Balazs, N.: Equiprobability, information and entropy in quantum theory. Ann. Phys. (NY) 179, 97–144 (1987)CrossRefMathSciNetGoogle Scholar
  95. 95.
    Balian, R., Alhassid, Y., Reinhardt, H.: Dissipation in many-body systems: a geometric approach based on information theory. Phys. Rep. 131, 1–146 (1986)CrossRefMathSciNetGoogle Scholar
  96. 96.
    Barbaresco F.: Information/contact geometries and Koszul entropy. In: Nielsen, F., Barbaresco, F. (eds.) Geometric Science of Information. Lecture Notes in Computer Science, vol. 8085, pp. 604–611. Springer, Berlin (2013)Google Scholar
  97. 97.
    Souriau J.M.: Définition covariante des équilibres thermodynamiques. Suppl. Nuovo Cimento 1, 203–216 (1966)Google Scholar
  98. 98.
    Souriau, J.M.: Thermodynamique et Géométrie 676, 369–397 (1978)MathSciNetGoogle Scholar
  99. 99.
    Souriau, J.M.: Géométrie de l’espace de phases. Commun. Math. Phys. 1, 374 (1966)Google Scholar
  100. 100.
    Souriau, J.M.: On geometric mechanics. Discrete Continuous Dyn. Syst. 19(3), 595–607 (2007)Google Scholar
  101. 101.
    Souriau, J.M.: Structure des Systèmes Dynamiques. Dunod, Paris (1970)Google Scholar
  102. 102.
    Souriau, J.M.: Structure of dynamical systems. Progress in Mathematics, vol. 149. Birkhäuser Boston Inc., Boston. A symplectic view of physics (translated from the French by Cushman-de Vries, C.H.) (1997)Google Scholar
  103. 103.
    Souriau, J.M.: Thermodynamique relativiste des fluides. Rend. Sem. Mat. Univ. e Politec. Torino, 35:21–34 (1978), 1976/77Google Scholar
  104. 104.
    Souriau, J.M., Iglesias, P.: Heat cold and geometry. In: Cahen, M., et al. (eds.) Differential Geometry and Mathematical Physics, pp. 37–68 (1983)Google Scholar
  105. 105.
    Souriau, J.M.: Thermodynamique et géométrie. In: Differential Geometrical Methods in Mathematical Physics, vol. 2 (Proceedings of the International Conference, University of Bonn, Bonn, 1977). Lecture Notes in Mathematics, vol. 676, pp. 369–397. Springer, Berlin (1978)Google Scholar
  106. 106.
    Souriau, J.M.,: Dynamic systems structure (Chap. 16 Convexité, Chap. 17 Mesures, Chap. 18 Etats Statistiques, Chap. 19 Thermodynamique), unpublished technical notes, available in Souriau archive (document sent by Vallée, C.)Google Scholar
  107. 107.
    Vallée, C.: Lois de comportement des milieux continus dissipatifs compatibles avec la physique relativiste, thèse, Poitiers University (1978)Google Scholar
  108. 108.
    Iglésias P., Equilibre statistiques et géométrie symplectique en relativité générale. Ann. l’Inst. Henri Poincaré, Sect. A, Tome 36(3), 257–270 (1982)Google Scholar
  109. 109.
    Iglésias, P.: Essai de thermodynamique rationnelle des milieux continus. Ann. l’Inst. Henri Poincaré, 34, 1–24 (1981)Google Scholar
  110. 110.
    Vallée, C.: Relativistic thermodynamics of continua. Int. J. Eng. Sci. 19(5), 589–601 (1981)Google Scholar
  111. 111.
    Pavlov, V.P., Sergeev, V.M.: Thermodynamics from the differential geometry standpoint. Theor. Math. Phys. 157(1), 1484–1490 (2008)Google Scholar
  112. 112.
    Kozlov, V.V.: Heat Equilibrium by Gibbs and Poincaré. RKhD, Moscow (2002)Google Scholar
  113. 113.
    Berezin, F.A.: Lectures on Statistical Physics. Nauka, Moscow (2007) (English trans., World Scientific, Singapore, 2007)Google Scholar
  114. 114.
    Poincaré, H.: Réflexions sur la théorie cinétique des gaz. J. Phys. Theor. Appl. 5, 369–403 (1906)CrossRefzbMATHGoogle Scholar
  115. 115.
    Carathéodory, C.: Math. Ann. 67, 355–386 (1909)CrossRefMathSciNetGoogle Scholar
  116. 116.
    Nakajima, S.: On quantum theory of transport phenomena. Prog. Theor. Phys. 20(6), 948–959 (1958)CrossRefzbMATHMathSciNetGoogle Scholar
  117. 117.
    Zwanzig, R.: Ensemble method in the theory of irreversibility. J. Chem. Phys. 33(5), 1338–1341 (1960)CrossRefMathSciNetGoogle Scholar
  118. 118.
    Bourdon, M.: Structure conforme au bord et flot géodésique d’un CAT(-1)-espace. L’Enseignement Math. 41, 63–102 (1995)Google Scholar
  119. 119.
    Viterbo, C.: Generating functions, symplectic geometry and applications. In: Proceedings of the International Congress Mathematics, Zurich (1994)Google Scholar
  120. 120.
    Viterbo, C.: Symplectic topology as the geometry of generating functions. Math. Ann. 292, 685–710 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  121. 121.
    Hörmander, L.: Fourier integral operators I. Acta Math. 127, 79–183 (1971)CrossRefzbMATHMathSciNetGoogle Scholar
  122. 122.
    Théret, D.: A complete proof of Viterbo’s uniqueness theorem on generating functions. Topology Appl. 96, 249–266 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  123. 123.
    Pansu, P.: Volume, courbure et entropie. Séminaire Bourbaki 823, 83–103 (1996)Google Scholar
  124. 124.
    Besson, G., Courtois, G., Gallot, S.: Entropies et rigidités des espaces localement symétriques de courbure strictement négative. Geom. Funct. Anal. 5, 731–799 (1995)Google Scholar
  125. 125.
    Fréchet, M.: Sur quelques points du calcul fonctionnel. Rend. Circolo Math. Palermo 22, 1–74 (1906)CrossRefzbMATHGoogle Scholar
  126. 126.
    Fréchet, M.: L’espace des courbes n’est qu’un semi-espace de Banach. General Topology and Its Relation to Modern Analysis and Algebra, pp. 155–156, Prague (1962)Google Scholar
  127. 127.
    Alt, H., Godau, M.: Computing the Fréchet distance between two polygonal curves. Int. J. Comput. Geom. Appl. 5, 75–91 (1995)Google Scholar
  128. 128.
    Fréchet, M.R.: Les éléments aléatoires de nature quelconque dans un espace distancié. Ann. l’Inst. Henri Poincaré 10(4), 215–310 (1948)Google Scholar
  129. 129.
    Marle, C.M.: On mechanical systems with a Lie group as configuration space. In: M. de Gosson (ed.) Jean Leray ’99 Conference Proceedings: the Karlskrona Conference in the Honor of Jean Leray, Kluwer, Dordrecht, pp. 183–203 (2003)Google Scholar
  130. 130.
    Marle, C.M.: On Henri Poincaré’s note “Sur une forme nouvelle des équations de la mécanique”. JGSP 29, 1–38 (2013)zbMATHMathSciNetGoogle Scholar
  131. 131.
    Kloeckner, B.: Géométrie des bords: compactifications différentiables et remplissages holomorphes. Thèse Ecole Normale Supérieure de Lyon. (2006). Accessed Dec 2006
  132. 132.
    Barbaresco, F.: Super Resolution Spectrum Analysis Regularization: Burg, Capon and Ago-antagonistic Algorithms, EUSIPCO-96, pp. 2005–2008, Trieste (1996)Google Scholar
  133. 133.
    Barbaresco, F.: Computation of most threatening radar trajectories areas and corridors based on fast-marching and level sets. In: IEEE CISDA Symposium, Paris (2011)Google Scholar
  134. 134.
    Michor, P.W., Mumford, D.: An overview of the Riemannian metrics on spaces of curves using the Hamiltonnian approach. Appl. Comput. Harm. Anal. 23(1), 74–113 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  135. 135.
    Chouakria-Douzal, A., Nagabhusha, P.N.: Improved Fréchet distance for time series. In: Data Sciences and Classification, pp. 13–20. Springer, Berlin (2006)Google Scholar
  136. 136.
    Bauer, M., et al.: Constructing reparametrization invariant metrics on spaces of plane curves. Preprint
  137. 137.
    Fréchet, M.: L’espace dont chaque élément est une courbe n’est qu’un semi-espace de Banach. Ann. Sci. l’ENS, 3ème série. Tome 78(3), 241–272 (1961)Google Scholar
  138. 138.
    Fréchet, M.: L’espace dont chaque élément est une courbe n’est qu’un semi-espace de Banach II. Ann.Sci. l’ENS, 3ème série. Tome 80(2), pp. 135–137 (1963)Google Scholar
  139. 139.
    Chazal, F., et al.: Gromov-Hausdorff stable signatures for shapes using persistence. In: Eurographics Symposium on Geometry Processing 2009, Marc Alexa and Michael Kazhdan (Guest editors), vol. 28, no. 5 (2009)Google Scholar
  140. 140.
    Cagliari, F., Di Fabio B., Landi, C.: The natural pseudo-distance as a quotient pseudo metric, and applications. Preprint
  141. 141.
    Frosini, P., Landi, C.: No embedding of the automorphisms of a topological space into a compact metric space endows them with a composition that passes to the limit. Appl. Math. Lett. 24(10), 1654–1657 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  142. 142.
    Shevchenko, O.: Recursive construction of optimal self-concordant barriers for homogeneous cones. J. Optim. Theor. Appl. 140(2), 339–354 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  143. 143.
    Güler, O., Tunçel, L.: Characterization of the barrier parameter of homogeneous convex cones. Math. Program. 81(1), Ser. A, 55–76 (1998)Google Scholar
  144. 144.
    Rothaus, O.S.: Domains of positivity. Bull. Am. Math. Soc. 64, 85–86 (1958)CrossRefzbMATHMathSciNetGoogle Scholar
  145. 145.
    Nesterov, Y., Nemirovskii, A.: Interior-point polynomial algorithms. In: Convex Programming, SIAM Studies in Applied Mathematics, vol. 13 (1994)Google Scholar
  146. 146.
    Vinberg, E.B.: The theory of homogeneous convex cones. Tr. Mosk. Mat. O-va. 12, 303–358 (1963)MathSciNetGoogle Scholar
  147. 147.
    Rothaus, O.S.: The construction of homogeneous convex cones. Ann. Math. Ser. 2, 83, 358–376 (1966)Google Scholar
  148. 148.
    Güler, O.: Barrier functions in interior point methods. Math. Oper. Res. 21(4), 860–885 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  149. 149.
    Uehlein, F.A.: Eidos and Eidetic Variation in Husserl’s Phenomenology. In: Language and Schizophrenia, Phenomenology, pp. 88–102. Springer, New York (1992)Google Scholar
  150. 150.
    Bergson, H.: L’évolution créatrice. Les Presses universitaires de France, Paris (1907).
  151. 151.
    Riquier, C.: Bergson lecteur de Platon: le temps et l’eidos, dans interprétations des idées platoniciennes dans la philosophie contemporaine (1850–1950), coll. Tradition de la pensée classique, Paris, Vrin (2011)Google Scholar
  152. 152.
    Worms, F.: Bergson entre Russel et Husserl: un troisième terme? In: Rue Descartes, no. 29, Sens et phénomène, philosophie analytique et phénoménologie, pp. 79–96, Presses Universitaires de France, Sept. 2000Google Scholar
  153. 153.
    Worms, F.: Le moment 1900 en philosophie. Presses Universitaires du Septentrion, premier trimestre, Etudes réunies sous la direction de Frédéric Worms (2004)Google Scholar
  154. 154.
    Worms, F.: Bergson ou Les deux sens de la vie: étude inédite, Paris, Presses universitaires de France, Quadrige. Essais, débats (2004)Google Scholar
  155. 155.
    Bergson, H., Poincaré, H.: Le matérialisme actuel. Bibliothèque de Philosophie Scientifique, Paris, Flammarion (1920)Google Scholar
  156. 156.
    de Saxcé G., Vallée C.: Bargmann group, momentum tensor and Galilean invariance of Clausius-Duhem inequality. Int. J. Eng. Sci. 50, 216–232 (2012)Google Scholar
  157. 157.
    Dubois, F.: Conservation laws invariants for Galileo group. CEMRACS preliminary results. ESAIM Proc. 10, 233–266 (2001)Google Scholar
  158. 158.
    Moreau, J.J.: Fonctions convexes duales et points proximaux dans un espace hilbertien. C. R. l’Acad. des Sci. Série A, Tome 255, 2897–2899 (1962)zbMATHMathSciNetGoogle Scholar
  159. 159.
    Nielsen, F.: Hypothesis testing, information divergence and computational geometry. In: GSI’13 Conference, Paris, pp. 241–248 (2013)Google Scholar
  160. 160.
    Vey, J.: Sur une notion d’hyperbolicité des variables localement plates. Faculté des sciences de l’université de Grenoble, Thèse de troisième cycle de mathématiques pures (1969)Google Scholar
  161. 161.
    Ruelle, D.: Statistical mechanics. In: Rigorous Results (Reprint of the 1989 edition). World Scientific Publishing Co., Inc, River Edge. Imperial College Press, London (1999)Google Scholar
  162. 162.
    Ruelle, D.: Hasard et Chaos. Editions Odile Jacob, Aout (1991)Google Scholar
  163. 163.
    Shima, H.: Geometry of Hessian Structures. In: Nielsen, F., Barbaresco, F. (eds.) Lecture Notes in Computer Science, vol. 8085, pp. 37–55. Springer, Berlin (2013)Google Scholar
  164. 164.
    Shima, H.: The Geometry of Hessian Structures. World Scientific, London (2007)Google Scholar
  165. 165.
    Zia, R.K.P., Redish Edward F., McKay Susan, R.: Making Sense of the Legendre Transform (2009), arXiv:0806.1147, June 2008
  166. 166.
    Fréchet, M.: Sur l’écart de deux courbes et sur les courbes limites. Trans. Am. Math. Soc. 6(4), 435–449 (1905)Google Scholar
  167. 167.
    Taylor, A.E., Dugac, P.: Quatre lettres de Lebesgue à Fréchet. Rev. d’Hist. Sci. Tome 34(2), 149–169 (1981)Google Scholar
  168. 168.
    Jensen, J.L.W.: Sur les fonctions convexes et les inégalités entre les valeurs moyennes. Acta Math. 30(1), 175–193 (1906)Google Scholar
  169. 169.
    Needham, T.: A visual explanation of Jensen’s inequality. Am. Math. Mon. 8, 768–77 (1993)Google Scholar
  170. 170.
    Donaldson, S.K.: Scalar curvature and stability of toric variety. J. Differ. Geom. 62, 289–349 (2002)Google Scholar
  171. 171.
    Abreu, M.: Kähler geometry of toric varieties and extremal metrics. Int. J. Math. 9, 641–651 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  172. 172.
    Atiyah, M., Bott, R.: The moment map and equivariant cohomology. Topology 23, 1–28 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  173. 173.
    Guan, D.: On modified Mabuchi functional and Mabuchi moduli space of kahler metrics on toric bundles. Math. Res. Lett. 6, 547–555 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  174. 174.
    Guillemin, V.: Kaehler structures on toric varieties. J. Differ. Geom. 40, 285–309 (1994)zbMATHMathSciNetGoogle Scholar
  175. 175.
    Guillemin, V.: Moment maps and combinatorial invariants of Hamiltonian T\(^{n}\)-spaces, Birkhauser (1994)Google Scholar
  176. 176.
    Crouzeix, J.P.: A relationship between the second derivatives of a convex function and of its conjugate. Math. Program. 3, 364–365 (1977) (North-Holland)Google Scholar
  177. 177.
    Seeger, A.: Second derivative of a convex function and of its Legendre-Fenchel transformate. SIAM J. Optim. 2(3), 405–424 (1992)Google Scholar
  178. 178.
    Hiriart-Urruty, J.B.: A new set-valued second-order derivative for convex functions. Mathematics for Optimization, Mathematical Studies, vol. 129. North Holland, Amsterdam (1986)Google Scholar
  179. 179.
    Berezin, F.: Lectures on Statistical Physics (Preprint 157). Max-Plank-Institut für Mathematik, Bonn (2006)Google Scholar
  180. 180.
    Hill, R., Rice, J.R.: Elastic potentials and the structure of inelastic constitutive laws. SIAM J. Appl. Math. 25(3), 448–461 (1973)Google Scholar
  181. 181.
    Bruguières, A.: Propriétés de convexité de l’application moment, séminaire N. Bourbaki, exp. no. 654, pp. 63–87 (1985–1986)Google Scholar
  182. 182.
    Condevaux, M., Dazord, P., Molino, P.: Géométrie du moment. Trav. Sémin. Sud-Rhodanien Géom. Univ. Lyon 1, 131–160 (1988)Google Scholar
  183. 183.
    Delzant, T.: Hamiltoniens périodiques et images convexes de l’application moment. Bull. Soc. Math. Fr. 116, 315–339 (1988)zbMATHMathSciNetGoogle Scholar
  184. 184.
    Guillemin, V., Sternberg, S.: Convexity properties of the moment mapping. Inv. Math. 67, 491–513 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  185. 185.
    Guillemin, V., Sternberg, S.: Convexity properties of the moment mapping. Inv. Math. 77, 533–546 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  186. 186.
    Kirwan, F.: Convexity properties of the moment mapping. Inv. Math. 77, 547–552 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  187. 187.
    Deza, E., Deza, M.M.: Dictionary of Distances. Elsevier, Amsterdam (2006)Google Scholar
  188. 188.
    Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. II 106(4), 620–630 (1957)CrossRefzbMATHMathSciNetGoogle Scholar
  189. 189.
    Jaynes, E.T.: Information theory and statistical mechanics II. Phys. Rev. II 108(2), 171–190 (1957)CrossRefMathSciNetGoogle Scholar
  190. 190.
    Jaynes, E.T.: Prior probabilities. IEEE Trans. Syst. Sci. Cybern. 4(3), 227–241 (1968)CrossRefzbMATHGoogle Scholar
  191. 191.
    Amari, S.I., Nagaoka, H.: Methods of Information Geometry (Translation of Mathematical Monographs), vol. 191. AMS, Oxford University Press, Oxford (2000)Google Scholar
  192. 192.
    Amari, S.I.: Differential Geometrical Methods in Statistics. Lecture Notes in Statistics, vol. 28. Springer, Berlin (1985)CrossRefzbMATHGoogle Scholar
  193. 193.
    Rao, C.R.: Information and the accuracy attainable in the estimation of statistical parameters. Bull. Calcutta Math. Soc. 37, 81–89 (1945)zbMATHMathSciNetGoogle Scholar
  194. 194.
    Chentsov, N.N.: Statistical decision rules and optimal inferences. In: Transactions of Mathematics Monograph, vol. 53. American Mathematical Society, Providence (1982) (Published in Russian in 1972)Google Scholar
  195. 195.
    Trouvé, A., Younes, L.: Diffeomorphic matching in 1d: designing and minimizing matching functionals. In: Vernon, D. (ed.) Proceedings of ECCV (2000)Google Scholar
  196. 196.
    Trouvé, A., Younes, L.: On a class of optimal matching problems in 1 dimension. SIAM J. Control Opt. 39(4), 1112–1135 (2001)CrossRefGoogle Scholar
  197. 197.
    Younes, L.: Computable elastic distances between shapes. SIAM J. Appl. Math 58, 565–586 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  198. 198.
    Younes, L.: Optimal matching between shapes via elastic deformations. Image Vis. Comput. 17, 381–389 (1999)Google Scholar
  199. 199.
    Younes, L., Michor, P.W., Shah, J., Mumford, D.: A metric on shape space with explicit geodesics. Rend. Lincei Mat. Appl. 9, 25–57 (2008)MathSciNetGoogle Scholar
  200. 200.
    Kapranov, M.: Thermodynamics and the Moment Map (preprint), arXiv:1108.3472, Aug 2011

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Authors and Affiliations

  1. 1.Thales Air SystemsLimoursFrance

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