Maarit Järvenpää

Porous measures on 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.

Packing dimension of mean porous measures

We prove that the packing dimension of any mean porous Radon measure on Rd may be estimated from above by a function which depends on mean porosity. The upper bound tends to d . 1 as mean porosity tends to its maximum value. This result was stated in D. B. Beliaev and S. K. Smirnov [�eOn dimension of porous measures�f, Math. Ann. 323 (2002) 123.141], and in a weaker form in E. J�Narvenp�Na�Na and M. J�Narvenp�Na�Na [�ePorous measures on Rn: local structure and dimensional properties�f, Proc. Amer. Math. Soc. (2) 130 (2002) 419.426], but the proofs are not correct. Quite surprisingly, it turns out that mean porous measures are not necessarily approximable by mean porous sets. We verify this by constructing an example of a mean porous measure �E on R such that �E(A) = 0 for all mean porous sets A �¼ R.

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 also investigate some examples of planar sets with Hausdorff dimension bigger than 1. In particular, we prove that for quasi-circles in the plane all visible parts have Hausdorff dimension equal to 1.

Hausdorff and packing dimensions and sections of measures

Let m and n be integers with 0 m n and let μ be a Radon measure on ℝ n with compact support. For the Hausdorff dimension, dim H , of sections of measures we have the following equality: for almost all ( n − m )-dimensional linear subspaces V provided that dim H μ > m . Here μ v,a is the sliced measure and V ⊥ is the orthogonal complement of V . If the ( m + d )-energy of the measure μ is finite for some d >0, then for almost all ( n − m )-dimensional linear subspaces V we have

Hausdorff dimension of affine random covering sets in torus

We calculate the almost sure Hausdorff dimension of the random covering set lim supn→∞(gn + ξn) in d-dimensional torus T, where the sets gn ⊂ T are parallelepipeds, or more generally, linear images of a set with nonempty interior, and ξn ∈ T are independent and uniformly distributed random points. The dimension formula, derived from the singular values of the linear mappings, holds provided that the sequences of the singular values are decreasing.

Measurability of equivalence classes and MEC

We prove that a locally compact metric space that supports a doubling measure and a weak p-Poincaré inequality for some 1 ≤ p < ∞ is a MECp-space. The methods developed for this purpose include measurability considerations and lead to interesting consequences. For example, we verify that each extended real valued function having a p-integrable upper gradient is locally p-integrable.

_p

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.

-property in metric spaces

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.

One-dimensional families of projections

Let

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

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