All Questions
8 questions with no upvoted or accepted answers
3
votes
0
answers
91
views
Asymptotical control of the measure of tubes covering subsets of fixed Hausdorff dimension
(A version of this question was posted on math stack exchange)
Let $M$ be a $C^1$ submanifold of dimension $n$ of $\mathbb{R}^N$, and denote $\mu$ the standard surface measure on $M$.
Consider a ...
- 1,035
3
votes
0
answers
247
views
The projection of density $1$ point on a rectifiable set
I posted this question on MSE but I received no reply. So I repost this here for better luck. Thank you!
Let $\Gamma\subset \mathbb R^N$ be $\mathcal H^{N-1}$-rectifiable. Then we know that $\mathcal ...
- 679
2
votes
0
answers
91
views
Measurability of the union of cut loci along a curve
Let $(M,g)$ be a Riemannian symmetric space and $\alpha(s)$ be a geodesic. Define
$$
U(t)=\cup_{s\in[0,t]}{\rm Cut}(\alpha(s))
$$
as the union of the cut loci ${\rm Cut}(\alpha(s))$ along the curve $\...
- 125
2
votes
0
answers
354
views
Continuity of surface integrals on level sets
Let $\phi:\mathbb{R}^2\to\mathbb{R}$ such that $\phi^{-1}(0)\neq\emptyset$ and $\phi\in C^1(W)$ where $W$ is a compact neighborhood of $\phi^{-1}(0)$, with $\nabla\phi\neq 0$ in $W$. So there is some $...
- 1,759
1
vote
0
answers
122
views
Differentiation under the integral sign for a $L^1$-valued function (shape derivative)
Let
$d\in\mathbb N$;
$U\subseteq\mathbb R^d$ be open and $$\mathcal A:=\{\Omega\subseteq U:\Omega\text{ is bounded and open and }\partial\Omega\text{ is of class }C^{0,\:1}\};$$
$E:=\bigcup_{\Omega\...
- 167
1
vote
1
answer
334
views
Relationship between volume density and area density
Let $\mu(x)dx$ be a measure in $\mathbb{R}^{2n-2}$, where $\mu$ (a $C^\infty$ and positive function) is the density of the volume in the sense that $\DeclareMathOperator{\Vol}{\mathrm{Vol}} \Vol_\mu(...
0
votes
0
answers
96
views
If $M$ is a manifold, $x∈M$ and $d(x,ω)=\inf\{t>0:x+tω∈M\}$, does the pushforward of the solid angle measure under $S^2∋ω↦x+d(x,ω)ω$ admit a density?
Let $S^2$ denote the unit 2-sphere, $M$ be a 2-dimensional oriented embedded $C^1$-submanifold of $\mathbb R^3$ with $$d_M(x,\omega):=\inf\left\{t>0:x+t\omega\in M\right\}<\infty\;\;\;\text{for ...
- 167
0
votes
0
answers
148
views
existence of locally translation-invariant Borel measure on Frechet manifolds
It is well known that the only locally finite, translation-invariant Borel measure on an
infinite-dimensional, separable Frechet space is the trivial measure. I am wondering about an analogous ...
