English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Open

Presents the pdf analysis as premium feature

PDF Analysis

Insights

Presents the summarisation as premium feature

Summarisation

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Our main result explains in what sense typical visible parts of a set with large Hausdorff dimension are smaller than the set itself. This is achieved by generalizing the notation of sliced measures by means of transversal mappings, and by establishing a connection between dimensional properties of generalized slices and those of visible parts. 1. Background and preliminary discussion Given integers k and d such that 0 ≤ k ≤ d−1, and an affine k-plane K in R (0-plane is simply a point), we use the notation ProjK for the projection onto K. The following definition of visibility goes back to Urysohn [U] in the 1920’s: Let E ⊂ R be compact. A point a ∈ E is visible from K, if a is the only point of E in the line segment joining a to ProjK(a). The visible part of E from K, denoted by VK(E), is the set of all points that are visible from K. Visibility was investigated in connection with set theoretic problems by Nikodym in [N]. The study of dimensional properties of visible parts, in turn, initiated in [JJMO]. Denoting by dimH the Hausdorff dimension, we have for almost all affine k-planes K not intersecting E that (1.1) dimH(VK(E)) = dimHE provided that dimHE ≤ d − 1 [JJMO, Theorem 3.2]. On the other hand, under the assumption dimHE > d− 1, we have (1.2) dimH(VK(E)) ≥ d− 1 for almost all k-planes K not intersecting E [JJMO, Proposition 3.3]. It is not known whether the opposite inequality holds in (1.2). Some special examples of planar sets with Hausdorff dimension bigger than 1 are investigated in [JJMO]. In particular, it is shown that Hausdorff dimensions of all visible parts of a quasi-circle are equal to 1. In [O] an upper bound bigger than 1 is verified for Hausdorff dimensions of 2000 Mathematics Subject Classification. 28A78, 28A80, 28A15.

Transversal Mappings between Manifolds and Non-Trivial Measures on Visible Parts