TRANSVERSAL MAPPINGS BETWEEN MANIFOLDS AND NON-TRIVIAL MEASURES ON VISIBLE PARTS

Abstract. 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 Rd (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 ⊂ Rd 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 o