Frequency of Sobolev and quasiconformal dimension distortion

We study Hausdorff and Minkowski dimension distortion for images of generic affine subspaces of Euclidean space under Sobolev and quasiconformal maps. For a supercritical Sobolev map f defined on a domain in R-n, we estimate from above the Hausdorff dimension of the set of affine subspaces parallel to a fixed in-dimensional linear subspace, whose image under f has positive H-alpha measure for some fixed alpha > m. As a consequence, we obtain new dimension distortion and absolute continuity statements valid for almost every affine subspace. Our results hold for mappings taking values in arbitrary metric spaces, yet are new even for quasiconformal maps of the plane. We illustrate our results with numerous examples. (C) 2012 Elsevier Masson SAS. All rights reserved.