We establish formulas for bounds on the Haudorff measure of the intersection of certain Cantor sets with their translates. As a consequence we obtain a formula for the Hausdorff dimensions of these intersections.

We construct quasiconformal mappings $f\colon \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ for which there is a Borel set $E \subset \mathbb{R}^2 \times \{0\}$ of positive Lebesgue $2$-measure whose image $f(E)$ has Hausdorff zero. This gives solution to the open problem inverse absolute continuity on hypersurfaces, attributed Gehring. By implication, our result also answers questions V\"ais\"al\...

In a recent paper, Pertti Mattila asked which gauge functions φ have the property that for any Borel set A ⊂ R2 with Hausdorff measure Hφ(A) > 0, the projection of A to almost every line has positive length. We show that finiteness of ∫ 1 0 φ(r) r2 dr, which is known to be sufficient for this property, is also necessary for regularly varying φ. Our proof is based on a random construction adapte...

We give a necessary and sufficient condition on a Randers space for the existence of a measure for which Shen’s S-curvature vanishes everywhere. Moreover, such a measure coincides with the Busemann-Hausdorff measure up to a constant multiplication.

In this article we describe recent (within the past 4 or 5 years) results in minimal surface theory, with particular emphasis on the existence and regularity theory. For a more general survey, including some technical background discussion and a more comprehensive bibliography, the reader is referred to [30]. Throughout the article «?f will denote ÄJ-dimensional Hausdorff measure (in Euclidean ...

In the present paper we sketch the proof of the fact that for any open connected set Ω ⊂ Rn+1, n ≥ 1, and any E ⊂ ∂Ω with 0 < H(E) < ∞, absolute continuity of the harmonic measure ω with respect to the Hausdorff measure on E implies that ω|E is rectifiable.

If X is a subset of R , let Hk(X) denote the k-dimensional hausdorff measure of X . We write Ik(X) = 0 if H(πKX) = 0 for almost every k-plane K in R , where πK : R → K is orthogonal projection. Otherwise we write Ik(X) > 0. This paper gives a proof of the following theorem. 1.1. Structure Theorem. Let X be a set in R with Hk(X) < ∞ and Ik(X) > 0. Then there is a k-dimensional C submanifold M wi...