Ioffe, Alexander (2006) *A Sard Theorem for Set-Valued Mappings.* [Preprint]

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## Abstract

If F is a set-valued mapping from IRn into IRm with closed graph, then y ∈ IRm is a critical value of F if for some x with y ∈ F(x), F is not metrically regular at (x, y). We prove that the set of critical values of a set-valued mapping whose graph is a definable (tame) set in an o-minimal structure containing additions and multiplications is a set of dimension not greater than m − 1 (resp. a porous set). As a corollary of this result we get that the collection of asymptotically critical values of a semialgebraic setvalued mapping has dimension not greater than m−1, thus extending to such mappings a corresponding result by Kurdyka-Orro-Simon for C1 semialgebraic mappings. Finally, as a by-product of the proof of the theorem, we get that a definable continuous real-valued function is constant on components of the set of its subdifferentiably critical points, thus extending to all definable functions a recent result of Bolte-Daniilidis-Lewis for globally subanalytic functions.

Item Type: | Preprint |
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Additional Information: | pubdom TRUE |

Uncontrolled Keywords: | o-minimal structure, definable set-valued mapping, rate of surjection, critical value |

Subjects: | 49-xx Calculus of variations and optimal control; optimization > 49Jxx Existence theories 58-xx Global analysis, analysis on manifolds > 58Kxx Theory of singularities and catastrophe theory 32-xx Several complex variables and analytic spaces > 32Bxx Local analytic geometry |

Faculty: | UNSPECIFIED |

Depositing User: | Users 1 not found. |

Date Deposited: | 18 Jan 2006 |

Last Modified: | 21 Apr 2010 11:13 |

URI: | https://docserver.carma.newcastle.edu.au/id/eprint/312 |

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