Continuity of set-valued maps revisited in the light of tame geometry

Details Continuity of set-valued maps revisited in the light of tame geometry Continuity of set-valued maps revisited in the light of tame geometry

2009-05-04

arxiv.org/abs/0905.0373v1

J. London Math. Soc. (2011) 83(3): 637-658

Mathematics

Optimization and Control

21 pages, 2 figures

Scientific paper

10.1112/jlms/jdq084

Continuity of set-valued maps is hereby revisited: after recalling some basic concepts of variational analysis and a short description of the State-of-the-Art, we obtain as by-product two Sard type results concerning local minima of scalar and vector valued functions. Our main result though, is inscribed in the framework of tame geometry, stating that a closed-valued semialgebraic set-valued map is almost everywhere continuous (in both topological and measure-theoretic sense). The result, depending on stratification techniques, holds true in a more general setting of o-minimal (or tame) set-valued maps. Some applications are briefly discussed at the end.

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