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Second order differentiability of convex functions in Banach spaces. (English) Zbl 0802.46027
Summary: We present a second order differentiability theory for convex functions on Banach spaces. Among others, we investigate whether an infinite dimensional version of A. D. Alexandrov’s classical result is valid. The latter states that a convex function on \(\mathbb{R}^ n\) is twice differentiable almost everywhere. We obtain partial positive answers for convex integral functionals.

MSC:
46B22 Radon-Nikodým, Kreĭn-Milman and related properties
26B25 Convexity of real functions of several variables, generalizations
46G05 Derivatives of functions in infinite-dimensional spaces
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[1] Ralph Abraham, Jerrold E. Marsden, and Tudor S. Raţiu, Manifolds, tensor analysis, and applications, Global Analysis Pure and Applied: Series B, vol. 2, Addison-Wesley Publishing Co., Reading, Mass., 1983. · Zbl 0508.58001
[2] A. D. Alexandroff, Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it, Leningrad State Univ. Annals [Uchenye Zapiski] Math. Ser. 6 (1939), 3 – 35 (Russian).
[3] N. Aronszajn, Differentiability of Lipschitzian mappings between Banach spaces, Studia Math. 57 (1976), no. 2, 147 – 190. · Zbl 0342.46034
[4] Hédy Attouch, Familles d’opérateurs maximaux monotones et mesurabilité, Ann. Mat. Pura Appl. (4) 120 (1979), 35 – 111 (French, with English summary). · Zbl 0416.47019 · doi:10.1007/BF02411939 · doi.org
[5] Hédy Attouch and Roger J.-B. Wets, Isometries for the Legendre-Fenchel transform, Trans. Amer. Math. Soc. 296 (1986), no. 1, 33 – 60. · Zbl 0607.49009
[6] Victor Bangert, Analytische Eigenschaften konvexer Funktionen auf Riemannschen Mannigfaltigkeiten, J. Reine Angew. Math. 307/308 (1979), 309 – 324 (German). · Zbl 0396.52007 · doi:10.1515/crll.1979.307-308.309 · doi.org
[7] A. Ben-Tal and J. Zowe, Directional derivatives in nonsmooth optimization, J. Optim. Theory Appl. 47 (1985), no. 4, 483 – 490. · Zbl 0556.90074 · doi:10.1007/BF00942193 · doi.org
[8] Jonathan Borwein, Simon Fitzpatrick, and Petar Kenderov, Minimal convex uscos and monotone operators on small sets, Canad. J. Math. 43 (1991), no. 3, 461 – 476. · Zbl 0746.46035 · doi:10.4153/CJM-1991-028-5 · doi.org
[9] J. M. Borwein and D. Preiss, A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions, Trans. Amer. Math. Soc. 303 (1987), no. 2, 517 – 527. · Zbl 0632.49008
[10] K. Buchner, Sard’s theorem on Banach manifolds. Topics in differential geometry. I, II, Colloq. Math. Soc. János Bolyai, vol. 46, North-Holland, Amsterdam, 1989.
[11] H. Busemann, Convex surfaces, Interscience, New York, 1955. · Zbl 0196.55101
[12] Herbert Busemann and Willy Feller, Krümmungseigenschaften Konvexer Flächen, Acta Math. 66 (1936), no. 1, 1 – 47 (German). · JFM 62.0832.02 · doi:10.1007/BF02546515 · doi.org
[13] Jens Peter Reus Christensen, Measure theoretic zero sets in infinite dimensional spaces and applications to differentiability of Lipschitz mappings, Publ. Dép. Math. (Lyon) 10 (1973), no. 2, 29 – 39. Actes du Deuxième Colloque d’Analyse Fonctionnelle de Bordeaux (Univ. Bordeaux, 1973), I, pp. 29 – 39. · Zbl 0302.43001
[14] V. F. Dem\(^{\prime}\)yanov, C. Lemaréchal, and J. Zowe, Approximation to a set-valued mapping. I. A proposal, Appl. Math. Optim. 14 (1986), no. 3, 203 – 214. · Zbl 0619.49005 · doi:10.1007/BF01442236 · doi.org
[15] J. Diestel and J. J. Uhl Jr., Vector measures, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis; Mathematical Surveys, No. 15. · Zbl 0369.46039
[16] M. Fabián, Lipschitz smooth points of convex functions and isomorphic characterizations of Hilbert spaces, Proc. London Math. Soc. (3) 51 (1985), no. 1, 113 – 126. · Zbl 0549.46025 · doi:10.1112/plms/s3-51.1.113 · doi.org
[17] Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. · Zbl 0176.00801
[18] Simon Fitzpatrick and R. R. Phelps, Differentiability of the metric projection in Hilbert space, Trans. Amer. Math. Soc. 270 (1982), no. 2, 483 – 501. · Zbl 0504.41029
[19] -, Bounded approximants to monotone operators in Banach spaces (to appear).
[20] J.-B. Hiriart-Urruty, Lipschitz \?-continuity of the approximate subdifferential of a convex function, Math. Scand. 47 (1980), no. 1, 123 – 134. · Zbl 0426.26005 · doi:10.7146/math.scand.a-11878 · doi.org
[21] J.-B. Hiriart-Urruty, Calculus rules on the approximate second-order directional derivative of a convex function, SIAM J. Control Optim. 22 (1984), no. 3, 381 – 404. · Zbl 0557.90077 · doi:10.1137/0322025 · doi.org
[22] J.-B. Hiriart-Urruty and A. Seeger, Calculus rules on a new set-valued second order derivative for convex functions, Nonlinear Anal. 13 (1989), no. 6, 721 – 738. · Zbl 0705.26013 · doi:10.1016/0362-546X(89)90090-4 · doi.org
[23] J.-B. Hiriart-Urruty and A. Seeger, The second-order subdifferential and the Dupin indicatrices of a nondifferentiable convex function, Proc. London Math. Soc. (3) 58 (1989), no. 2, 351 – 365. · Zbl 0632.53009 · doi:10.1112/plms/s3-58.2.351 · doi.org
[24] -, Complément de Schur et sous-différentiel du second ordre d’une fonction convexe (to appear). · Zbl 0748.49013
[25] Nobuyuki Kato, On the second derivatives of convex functions on Hilbert spaces, Proc. Amer. Math. Soc. 106 (1989), no. 3, 697 – 705. · Zbl 0671.47041
[26] F. Mignot, Contrôle dans les inéquations variationelles elliptiques, J. Functional Analysis 22 (1976), no. 2, 130 – 185 (French). · Zbl 0364.49003
[27] Jean-Jacques Moreau, Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France 93 (1965), 273 – 299 (French). · Zbl 0136.12101
[28] Umberto Mosco, Convergence of convex sets and of solutions of variational inequalities, Advances in Math. 3 (1969), 510 – 585. · Zbl 0192.49101 · doi:10.1016/0001-8708(69)90009-7 · doi.org
[29] A. S. Nemirovskiĭ, The polynomial approximation of functions on Hilbert space, Funkcional. Anal. i Priložen. 7 (1973), no. 4, 88 – 89 (Russian). A. S. Nemirovskiĭ and S. M. Semenov, The polynomial approximation of functions on Hilbert space, Mat. Sb. (N.S.) 92(134) (1973), 257 – 281, 344 (Russian). A. S. Nemirovskiĭ, The uniform approximation of Lipschitzian mappings of Banach spaces by Gâteaux differentiable mappings, Trudy Kaf. Teorii Funkciĭ i Funkcional. Anal. Moskov. Gos. Univ. 1 (1974), 12 – 16 (Russian). Analysis in infinite-dimensional spaces.
[30] Dominikus Noll, Second order differentiability of integral functionals on Sobolev spaces and \?²-spaces, J. Reine Angew. Math. 436 (1993), 1 – 17. · Zbl 0767.49010 · doi:10.1515/crll.1993.436.1 · doi.org
[31] Dominikus Noll, Generic Gâteaux-differentiability of convex functions on small sets, J. Math. Anal. Appl. 147 (1990), no. 2, 531 – 544. · Zbl 0715.46019 · doi:10.1016/0022-247X(90)90368-P · doi.org
[32] Dominikus Noll, Generic Fréchet-differentiability of convex functions on small sets, Arch. Math. (Basel) 54 (1990), no. 5, 487 – 492. · Zbl 0666.46049 · doi:10.1007/BF01188676 · doi.org
[33] Dominikus Noll, Generalized second fundamental form for Lipschitzian hypersurfaces by way of second epi derivatives, Canad. Math. Bull. 35 (1992), no. 4, 523 – 536. · Zbl 0764.53007 · doi:10.4153/CMB-1992-069-5 · doi.org
[34] R. R. Phelps, Gaussian null sets and differentiability of Lipschitz map on Banach spaces, Pacific J. Math. 77 (1978), no. 2, 523 – 531. · Zbl 0396.46041
[35] Robert R. Phelps, Convex functions, monotone operators and differentiability, Lecture Notes in Mathematics, vol. 1364, Springer-Verlag, Berlin, 1989. · Zbl 0658.46035
[36] D. Preiss, Differentiability of Lipschitz functions on Banach spaces, J. Funct. Anal. 91 (1990), no. 2, 312 – 345. · Zbl 0711.46036 · doi:10.1016/0022-1236(90)90147-D · doi.org
[37] R. T. Rockafellar, Maximal monotone relations and the second derivatives of nonsmooth functions, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), no. 3, 167 – 184 (English, with French summary). · Zbl 0581.49009
[38] R. T. Rockafellar, Generalized second derivatives of convex functions and saddle functions, Trans. Amer. Math. Soc. 322 (1990), no. 1, 51 – 77. · Zbl 0712.49011
[39] -, Conjugate duality and optimization, SIAM, Philadelphia, Pa., 1974. · Zbl 0296.90036
[40] R. Tyrrell Rockafellar, Second-order optimality conditions in nonlinear programming obtained by way of epi-derivatives, Math. Oper. Res. 14 (1989), no. 3, 462 – 484. · Zbl 0698.90070 · doi:10.1287/moor.14.3.462 · doi.org
[41] Gabriella Salinetti and Roger J.-B. Wets, On the relations between two types of convergence for convex functions, J. Math. Anal. Appl. 60 (1977), no. 1, 211 – 226. · Zbl 0359.54005 · doi:10.1016/0022-247X(77)90060-9 · doi.org
[42] Josef Stoer and Christoph Witzgall, Convexity and optimization in finite dimensions. I, Die Grundlehren der mathematischen Wissenschaften, Band 163, Springer-Verlag, New York-Berlin, 1970. · Zbl 0203.52203
[43] J. Vanderwerff, Smooth approximation in Banach spaces (to appear). · Zbl 0812.46005
[44] R. A. Wijsman, Convergence of sequences of convex sets, cones and functions. II, Trans. Amer. Math. Soc. 123 (1966), 32 – 45. · Zbl 0146.18204
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