Unavoidable Porous Sets and Nondifferentiable Maps
Unavoidable Porous Sets and Nondifferentiable Maps
This chapter discusses Γₙ-nullness of sets porous “¹at infinity” and/or existence of many points of Fréchet differentiability of Lipschitz maps into n-dimensional spaces. The results reveal a σ-porous set whose complement is null on all n-dimensional surfaces and the multidimensional mean value estimates fail even for ε-Fréchet derivatives. Previous chapters have established conditions on a Banach space X under which porous sets in X are Γₙ-null and/or the the multidimensional mean value estimates for Fréchet derivatives of Lipschitz maps into n-dimensional spaces hold. This chapter investigates in what sense the assumptions of these main results are close to being optimal.
Keywords: multidimensional mean value, Fréchet differentiability, Lipschitz map, ε-Fréchet derivative, Banach space, porous sets, Fréchet derivative
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