Results 1 to 10 of about 22 (21)
On sets with unit Hausdorff density in homogeneous groups
Forum of Mathematics, Sigma, 2023 It is a longstanding conjecture that given a subset E of a metric space, if E has unit $\mathscr {H}^{\alpha }\llcorner E$ -density almost everywhere, then E is an $\alpha $ -rectifiable set. We prove this conjecture under the assumption that
Antoine Julia, Andrea Merlo
doaj +1 more source
Minimising Hausdorff dimension under Hölder equivalence
Journal of the London Mathematical Society, Volume 103, Issue 3, Page 781-816, April 2021., 2021 Abstract We study the infimal value of the Hausdorff dimension of spaces that are Hölder equivalent to a given metric space; we call this bi‐Hölder‐invariant ‘Hölder dimension’. This definition and some of our methods are analogous to those used in the study of conformal dimension.
Samuel Colvin
wiley +1 more source
A Simple Proof of Dvoretzky-Type Theorem for Hausdorff Dimension in Doubling Spaces
Analysis and Geometry in Metric Spaces, 2022 The ultrametric skeleton theorem [Mendel, Naor 2013] implies, among other things, the following nonlinear Dvoretzky-type theorem for Hausdorff dimension: For any 0 < β < α, any compact metric space X of Hausdorff dimension α contains a subset which is ...
Mendel Manor
doaj +1 more source
CUBE PACKINGS IN EUCLIDEAN SPACES
Mathematika, Volume 67, Issue 2, Page 288-295, April 2021., 2021 Abstract In this paper, we study some cube packing problems. In particular, we are interested in compact subsets of Rn,n⩾2, which contain boundaries of cubes with all side lengths in (0,1). We show here that such sets must have lower box dimension at least n−0.5, and we will also provide sharp examples. We also show here that such sets must be large in
Han Yu
wiley +1 more source
Exceptional families of measures on Carnot groups
Analysis and Geometry in Metric Spaces, 2023 We study the families of measures on Carnot groups that have vanishing pp-module, which we call Mp{M}_{p}-exceptional families. We found necessary and sufficient Conditions for the family of intrinsic Lipschitz surfaces passing through a common point to ...
Franchi Bruno, Markina Irina
doaj +1 more source
Tangent points of lower content d‐regular sets and β numbers
Journal of the London Mathematical Society, Volume 101, Issue 2, Page 530-555, April 2020., 2020 Abstract Given a lower content d‐regular set in Rn, we prove that the subset of points in E where a certain Dini‐type condition on the so‐called Jones β numbers holds coincides with the set of tangent points of E, up to a set of Hd‐measure zero. The main point of our result is that Hd|E is not σ‐finite; because of this, we use a certain variant of the ...
Michele Villa
wiley +1 more source
Dimension Distortion by Sobolev Mappings in Foliated Metric Spaces
Analysis and Geometry in Metric Spaces, 2013 We quantify the extent to which a supercritical Sobolev mapping can increase the dimension of subsets of its domain, in the setting of metric measure spaces supporting a Poincaré inequality. We show that the set of mappings that distort the dimensions of
Balogh Zoltán M. +2 more
doaj +1 more source
International Journal of Mathematics and Mathematical Sciences, Volume 18, Issue 1, Page 97-106, 1995., 1993 We define the coordinate d‐dimension print to distinguish sets of same fractal dimension, and investigate its geometrical properties.
Hung Hwan Lee, In Soo Baek
wiley +1 more source
BiLipschitz Decomposition of Lipschitz Maps between Carnot Groups
Analysis and Geometry in Metric Spaces, 2015 Let f : G → H be a Lipschitz map between two Carnot groups. We show that if B is a ball of G, then there exists a subset Z ⊂ B, whose image in H under f has small Hausdorff content, such that B\Z can be decomposed into a controlled number of pieces, the ...
Li Sean
doaj +1 more source
Trace and observability inequalities for Laplace eigenfunctions on the torus
Forum of Mathematics, SigmaWe investigate trace and observability inequalities for Laplace eigenfunctions on the d-dimensional torus $\mathbb {T}^d$ , with respect to arbitrary Borel measures $\mu $ .
Nicolas Burq +3 more
doaj +1 more source