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Radiality and semismoothness

Jourani A.

Control and Cybernetics

2007

Vol. 36, no 3

669-680

We provide sufficient conditions for radiality and semismoothness. In general Banach spaces, we show that calmness ensures Dini-radiality as well as Dini-convexity of solution set to inequality systems. In finite dimensional spaces, we introduce the concept of Clarke-radiality and semismoothness of order m and show that each subanalytic set satisfies these properties. Similar properties are obtained for locally Lipschitzian subanalytic functions.

Interior distance of two points in a fiber of a subanalytic map

Kocel-Cynk B.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

Vol. 48, no 2

149-155

The aim of the paper is to prove the following theorem: Let N and M be two analytic manifolds and let K [is subset of] N be a compact subanalytic subset. For any continuous subanalytic map f : K --> M there exists a constant C > 0 such that for any y [belongs to] f(K) and any two points p, q in the same connected component of the fiber [f^-1](y) there exists a subanaiytic curve joining p and q in [f^-1](y) of length less than C.