Sets of minimal Hausdorff dimension for quasiconformal maps

For any 1 less than or equal to alpha less than or equal to n, there is a compact set E subset of R-n of (Hausdorff) dimension alpha whose dimension cannot be lowered by any quasiconformal map f : R-n --> R-n. We conjecture that no such set exists in the case alpha < 1. More generally, we identify a broad class of metric spaces whose Hausdorff dimension is minimal among quasisymmetric images.