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Results tagged with hausdorff-measure
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user 36721
Questions about Hausdorff measures, their variants (such as spherical Hausdorff measures) and generalisations.
4
votes
Accepted
If $\mathcal{H}^{n-1}(K)=0$ then $\mathcal{H}^n(K\times \mathbb{R})=0$
$\newcommand\de\delta\newcommand\ep\varepsilon\newcommand\R{\mathbb R}\newcommand\N{\mathbb N}$Saúl RM proved the desired result by an application of Frostman's lemma.
Here is an elementary proof:
Tak …
- 128k
1
vote
Extending a $C^1$ function on $\mathbb R^n$ to a set of finite $\mathcal H^{n-2}$ measure
This is not an answer, but rather a suggestion of an idea on how a possible counterexample could be constructed (if it exists).
Let $n=1$ (rather than $n\ge2$ as in the OP). Let $E$ be the Cantor subs …
- 128k
1
vote
Defining a measure of uniformity for measurable subsets of $[0,1]^2$ w.r.t dimension $\alpha...
Responding to the latest comment by the OP: "How would you suggest measuring the uniformity of measurable subsets of the unit square?" :
I think the idea of uniformity has hardly anything to do with …
- 128k
2
votes
Accepted
Defining a measure of uniformity for measurable subsets of $[0,1]^2$ w.r.t dimension $\alpha...
Here is another possible approach, perhaps closer to what the OP had in mind.
Let $S:=[0,1]^2$ be the unit square. "Partition" $S$ naturally into four congruent squares $S_{1,j}$ (with side length $1/ …
- 128k