We construct a Sierpinski carpet E ⊂ R of area zero and a K0 > 1 with the property that every K0-quasiconformal image of E contains rectifiable curves, but such that E has some quasiconformal image containing no non-constant rectifiable curves. 1991 Mathematics Subject Classification. Primary: 30C62 Secondary:

The study of uniformly distributed measures was crucial in Preiss’ proof of his theorem on rectifiability of measures with positive density. It is known that the support of a uniformly distributed measure is an analytic variety. In this paper, we provide quantitative information on the rectifiability of this variety. Tolsa had already shown that n-uniform measures are uniformly rectifiable. Her...

We prove that in any Banach space, the set of windows which a rectifiable curve resembles two or more straight line segments is quantitatively small with constants are independent curve, dimension and choice norm. Together Part I (also published this issue), completes proof necessary half analyst’s traveling salesman theorem sharp exponent uniformly convex spaces.

In this article we study some geometric properties of proximally smooth sets. First, introduce a modification the metric projection and prove its existence. Then provide an algorithm for constructing rectifiable curve between two sufficiently close points set in uniformly convex Banach space, with moduli smoothness convexity power type. Our returns reasonably short set, is iterative uses our pr...

We construct explicit examples of Frostman-type measures concentrated on arbitrary simple rectifiable curves \(\Gamma\subset\mathbb{R}^{2}\) positive length. Based such constructions we obtain for each \(p \in (1,\infty)\) an exact description the trace space \(W^{1}_{p}(\mathbb{R}^{2})|_{\Gamma}\) first-order Sobolev \(W^{1}_{p}(\mathbb{R}^{2})\) to curve \(\Gamma\) length.

We study the boundedness of the Hausdorff measure of the singular set of any solution for a semi-linear elliptic equation in general dimensional Euclidean space Rn. In our previous paper, we have clarified the structures of the nodal set and singular set of a solution for the semi-linear elliptic equation. In particular, we showed that the singular set is (n − 2)-rectifiable. In this paper, we ...

Let A be a compact set in Rp of Hausdorff dimension d. For s ∈ (0, d), the Riesz s-equilibrium measure μs is the unique Borel probability measure with support in A that minimizes Is(μ) := " 1 |x − y|s dμ(y)dμ(x) over all such probability measures. If A is strongly (Hd , d)-rectifiable, then μs converges in the weak-star topology to normalized d-dimensional Hausdorff measure restricted to A as s...

We describe analytic pencils of conics passing through the origin in C that can be mapped to straight lines locally near the origin by an analytic diffeomorphism. Under a minor non-degeneracy assumption, we prove that in a pencil with this property, almost all conics have 3 points of tangency with the same algebraic curve of class 3 (i. e. a curve projectively dual to a cubic). 2000 Math. Subj....