A Sard theorem for tame set-valued mappings |
| |
Authors: | A. Ioffe |
| |
Affiliation: | Department of Mathematics, Technion, Haifa 32000, Israel |
| |
Abstract: | If F is a set-valued mapping from Rn into Rm with closed graph, then y∈Rm 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 (respectively a σ-porous set). As a corollary of this result we get that the collection of asymptotically critical values of a set-valued mapping with a semialgebraic graph has dimension not greater than m−1. We also give an independent proof of the fact that a definable continuous real-valued function is constant on components of the set of its subdifferentiably critical points. |
| |
Keywords: | o-Minimal structure Definable set-valued mapping Rate of surjection Critical value |
本文献已被 ScienceDirect 等数据库收录! |
|