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Results tagged with hausdorff-measure
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user 61629
Questions about Hausdorff measures, their variants (such as spherical Hausdorff measures) and generalisations.
5
votes
2
answers
1k
views
Continuity of Hausdorff measure on level sets
Let $\Omega\subset\mathbb{R}^2$ a open and bounded set with smooth boundary and $\phi:\Omega\to\mathbb{R}$ a smooth function such that:
$\bullet$ $\phi^{-1}(0)\neq\emptyset$;
$\bullet$ $\nabla\phi(x)\ …
- 1,759
1
vote
1
answer
423
views
Why is the Hausdorff measure of this set zero?
Let $\Omega\subseteq\mathbb{R}^N$ be an open and bounded set, and let $\phi:\Omega\to\mathbb{R}^N$ be a $C^1$ function with the property that $\phi^{-1}(0)\neq\emptyset$, and $\nabla\phi(x)\neq 0,\ \f …
- 1,759
0
votes
0
answers
210
views
Signed distance function
Let $\Omega\subseteq\mathbb{R}^N$ be an open and bounded set with uniform Lipschitz boundary. Consider the signed distance function:
$d:\mathbb{R}^N\to\mathbb{R},\ d(x)=\begin{cases} \mathrm{dist}(x,\ …
- 1,759