Surfaces meeting porous sets in positive measure

Let X be a Banach space and 2 < n < dimX. We show there exists a directionally porous set P in X for which the set of C ¹ surfaces of dimension n meeting P in positive measure is not meager. If X is separable, this leads to a decomposition of X into the union of a σ-directionally porous set and a set which is null on residually many C ¹ surfaces of dimension n. This is of interest in the study of Γn-null and Γ-null sets and their applications to differentiability of Lipschitz functions.