Visible parts and dimensions

We study the visible parts of subsets of n-dimensional Euclidean space: a point a of a compact set A is visible from an affine subspace K of n, if the line segment joining PK(a) to a only intersects A at a (here PK denotes projection onto K). The set of all such points visible from a given subspace K is called the visible part of A from K. We prove that if the Hausdorff dimension of a compact set is at most n−1, then the Hausdorff dimension of a visible part is almost surely equal to the Hausdorff dimension of the set. On the other hand, provided that the set has Hausdorff dimension larger than n−1, we have the almost sure lower bound n−1 for the Hausdorff dimensions of visible parts. We al…

One-dimensional families of projections

Let m and n be integers with 0 < m < n. We consider the question of how much the Hausdorff dimension of a measure may decrease under typical orthogonal projections from onto m-planes provided that the dimension of the parameter space is one. We verify the best possible lower bound for the dimension drop and illustrate the sharpness of our results by examples. The question stems naturally from the study of measures which are invariant under the geodesic flow.

Local dimensions of sliced measures and stability of packing dimensions of sections of sets

Abstract Let m and n be integers with 0 R n to certain properties of plane sections of μ. This leads us to prove, among other things, that the lower local dimension of (n−m)-plane sections of μ is typically constant provided that the Hausdorff dimension of μ is greater than m. The analogous result holds for the upper local dimension if μ has finite t-energy for some t>m. We also give a sufficient condition for stability of packing dimensions of section of sets.

Dimensions of random affine code tree fractals

We calculate the almost sure Hausdorff dimension for a general class of random affine planar code tree fractals. The set of probability measures describing the randomness includes natural measures in random $V$-variable and homogeneous Markov constructions.

Porosities and dimensions of measures

We introduce a concept of porosity for measures and study relations between dimensions and porosities for two classes of measures: measures on $R^n$ which satisfy the doubling condition and strongly porous measures on $R$.

Porous measures on $\mathbb {R}^{n}$: Local structure and dimensional properties

We study dimensional properties of porous measures on R n . As a corollary of a theorem describing the local structure of nearly uniformly porous measures we prove that the packing dimension of any Radon measure on R n has an upper bound depending on porosity. This upper bound tends to n - 1 as porosity tends to its maximum value.

Visible parts of fractal percolation

We study dimensional properties of visible parts of fractal percolation in the plane. Provided that the dimension of the fractal percolation is at least 1, we show that, conditioned on non-extinction, almost surely all visible parts from lines are 1-dimensional. Furthermore, almost all of them have positive and finite Hausdorff measure. We also verify analogous results for visible parts from points. These results are motivated by an open problem on the dimensions of visible parts.

Packing dimension, intersection measures, and isometries