Surfaces meeting porous sets in positive measure (Q375895)

The author proves that any Banach space \(X\) of dimension \(\dim X>n>2\) contains a directionally porous set \(P\) for which the set of \(C^1\)-smooth \(n\)-dimensional surfaces which intersect \(P\) by a set of positive \(n\)-dimensional Lebesgue measures is of the second category. In the case where the space \(X\) is separable, this implies that \(X\) can be decomposed into a union of a sigma-directionally porous sets and a \(\Gamma_n\)-null set. This is in contrast to the case \(n=1,2\), when every sigma-directionally porous subset of a separable Banach space is \(\Gamma_1\)- and \(\Gamma_2\)-null and the case \(\dim X\leq n\), when \(\Gamma_n\)-null sets coincide with the Lebesgue null sets. By showing that every space of dimension greater than \(n\geq3\) contains porous sets which are not \(\Gamma_n\)-null, the paper also answers a question posed by \textit{J. Lindenstrauss} et al. [Fréchet differentiability of Lipschitz functions and porous sets in Banach spaces. Princeton, NJ: Princeton University Press (2012; Zbl 1241.26001)]. This connects the result of the present paper with \(\epsilon\)-differentiability of Lipschitz maps from \(X\) into spaces of dimension not exceeding \(n\).