Faculty of Mathematics and Computer Science

Differentiability for convex functions |

Code |
Semes-ter |
Hours: C+S+L |
Credits |
Type |
Section |

Teaching Staff in Charge |

Assoc.Prof. ANISIU Valer, Ph.D., anisiu@math.ubbcluj.ro |

Aims |

The study of convex functions defined on Banach or topological vector spaces, with emphasis on generic differentiability and related problemes. |

Content |

1. Convex functions defined on intervals.
Characterizations for convexity. The existence of one sided derivatives. Integral representation. 2. The continuity of convex functions. Upper and lower semicontinuity. Epigraph. Continuity and local boundedness, local Lipschitz property. Duality. 3. Gateaux differentiability. One sided derivatives, subgradients, Mazur's theorem on generic differentiability. The subdifferential as a monotone operator. 4. Frechet differentiability. Characterization of differentiability via the subdifferential. The Asplund-Preiss theorem on generic differentiability in Banach spaces with separable duals (using porosity). Asplund spaces, geometric characterizations. 5. Ekeland's theorem and applications. Ekeland's theorem. The differentiability of the norm in a Banach space. The theorems of Brondsted-Rockafellar, Bishop-Phelps. Applications to the geometry of Banach spaces. 6. Monotone operators. Upper semicontinuity, Kenderov's theorem on generic continuity. Applications to the differentiability of convex functions. The case of the domain with empty interior. |

References |

1. R.R. Phelps: Convex Functions, Monotone Operators and Differentiability. 2nd edition. Lecture Notes in Mathematics No.1364. Springer-Verlag 1993.
2. J.R. Giles: Convex analysis with applications in differentiation of convex functions. Res. Notes in Math. No.58, Pitman, Boston-London-Melbourne (1982) 3. D.G. De Figueiredo: The Ekeland variational principle with applications and detours. Tata Institute of fundamental research/Springer-Verlag 1989. 4. T. Precupanu: Spatii liniare topologice si elemente de analiza convexa. Ed. Academiei Romane, Bucuresti 1992. 5. Y. Benyamini, J. Lindenstrauss: Geometric Functional Analysis. American Math. Soc., 2000. |

Assessment |

Projects and exam. |