Sets of minimal Hausdorff dimension for quasiconformal maps

Abstract

For any 1 ≤ α ≤ n 1\le \alpha \le n , there is a compact set E ⊂ R n E\subset \mathbb {R}^n of (Hausdorff) dimension α \alpha whose dimension cannot be lowered by any quasiconformal map f : R n → R n f:\mathbb {R}^n\to \mathbb {R}^n . We conjecture that no such set exists in the case α > 1 \alpha >1 . More generally, we identify a broad class of metric spaces whose Hausdorff dimension is minimal among quasisymmetric images.