**3**

votes

**0**answers

28 views

### continuous injective extension of a map defined on a hemisphere

Let $S^2 = \{x\in \mathbf R^3\colon |x|=1\}$ be the unit sphere, $S^2_+ = S^2 \cap \{x_3 \ge 0\}$ be the upper hemisphere and $S^1 = S^2 \cap \{x_3 = 0\}$ be the unit circle. Let $u\colon S^2_+\to \...

**0**

votes

**1**answer

98 views

### Doubling metrics, doubling measures, Lebesgue density

As stated in this question,
Lebesgue differentiation theorem holds on locally doubling space?
and proved here,
http://www.math.uiuc.edu/~tyson/595f15lecture2.pdf
the Lebesgue differentiation theorem (...

**1**

vote

**0**answers

45 views

### BV functions with values in metric space

$
\newcommand{\IR}{\mathbb{R}}
\newcommand{\IN}{\mathbb{N}}
\newcommand{\supp}{\operatorname{supp}}
\newcommand{\divergence}{\operatorname{div}}
\newcommand{\Lip}{\operatorname{Lip}}
$
Let $E$ be a ...

**7**

votes

**1**answer

154 views

### Geometric Construct for Integrating Symmetric Tensors?

I'm interested in finding the appropriate geometric construct for the integration of symmetric tensors, analogous to the way differential forms can be integrated over manifolds.
The motivation comes ...

**1**

vote

**2**answers

43 views

### Polar coordinates, bounded domain with $C^{1}$ boundary

I have a question about a integral on a surface.
It is well known that for any Integrable function $f$ defined on $\mathbb{R}^{n}$, it holds that
\begin{equation}
(1) \quad \frac{d}{dr} \int_{B(0,r)}...

**1**

vote

**0**answers

74 views

### Differntiability of Distance to a CLosed Convex Set

Let $A$ be a closed convex set in Banach space $( \mathbb{R}^n, \| \cdot\| )$. For any $\mathbf{x} \in \mathbb{R}^n$, define $$Ｐ_{A}(\mathbf{x}) = \arg\min_{\mathbf{y}\in A} \| \mathbf{x} - \mathbf{y} ...

**1**

vote

**0**answers

77 views

### Can we always extract a proper Hausdorff measurable subset from a Hausdorff measurable set?

I also put this question on MSE here
Let $\Gamma\subset \Omega\subset \mathbb R^N$ be such that $\mathcal H^{N-1}(\Gamma)<+\infty$ (this also implise that $\Gamma$ is Hausdorff measurable).
Let $\...

**1**

vote

**1**answer

98 views

### Results for Hausdorff Measure after Linear Transformation

For the Sierpinski Triangle, $S$, the $d$ dimensional Hausdorff measure is given by, $H^{d}(S)$. If a linear transformation, $W$ is applied to $S$, with
$$W(x,y)=\begin{bmatrix} 1/2 & 0 \\ 0 &...

**2**

votes

**1**answer

78 views

### harmonic differential form integer class

Let $(M,g)$ be a compact Riemannian three-fold such that $H_2(M,\mathbb{Z}) = \mathbb{Z}$ and $S$ any surface representing 1. By Hodge theory there exist a harmonic differential one-form $\eta$ dual ...

**4**

votes

**1**answer

76 views

### sequence of graphs converge in the sense of varifold to multiplicity 2 plane

Say in $R^3$, is there a sequence of smooth graphs $f_n$ over some plane P, such that the graphs as submanifolds in $R^3$ converge in the sense of varifold (as Radon measures on $R^3 \times Gr(2,3)$ ) ...

**2**

votes

**0**answers

35 views

### Compute Mixed Volume with Respect to Some Regular Sets

Let $( \mathbb{R}^n, \mathcal{B}, \gamma)$ be a measure space where $\mathcal{B}$ is the Borel sigma algebra and $\gamma$ is a continuous measure. For $A, B\in \mathcal{B}$ that are convex, the mixed ...

**1**

vote

**0**answers

42 views

### Equivalent Definitions of the Gaussian Surface Measure for Regular Sets

I wonder if the following definitions of the Gaussian surface measure are equivalent.
First, let $\mathbb{R}^n$ be the Euclidean space and $A \subseteq \mathbb{R}^n$ be a sufficiently regular set, e....

**5**

votes

**0**answers

180 views

### Topology on the space of Borel measures

Let $ B $ be the set of all measures $ \phi $ of $ \mathbf{R}^{n} $ such that every open set is $ \phi $-measurable (sometimes these measures are called Borel measures). Note the measures in $ B $ are ...

**0**

votes

**0**answers

47 views

### The density one properties of $\mathcal H^{N-1}$-rectifiable set

Let $S\subset \mathbb R^N$ be a $\mathcal H^{N-1}$-rectifiable set. Then we know that there exist countably many Lipschitz $N-1$-graphs $\Gamma_i\subset \mathbb R^N$ such that
$$
\mathcal H^{N-1}\...

**7**

votes

**1**answer

277 views

### Level sets of weakly differentiable funtions

Let $C$ be a $C^1$ hypersurface in $R^n$ and let $u \in C^1(R^n)$. Suppose
$$\nabla u(x) \cdot \eta(x)=|\nabla u| \ \ \forall x\in C$$
where $\eta(x)$ is the normal vector to $C$ at $x$ ($\nabla u$ ...

**2**

votes

**0**answers

115 views

### question about currents

I have a question in the field of currents:
Let M be an n-dimensional smooth manifold, and let T be a k-current (induced by a k dimensional sub-manifold), I would like to approximate it by a series of ...

**5**

votes

**0**answers

69 views

### Does the Hodge *-operator act on the tangent space at 0 to the space of integral (n-1)-cycles in a conformal manifold of dimension d=2n?

Suppose $M$ is a compact, oriented conformal manifold of even dimension $d=2n$.
Write ${\cal D}^{\mathit{int}}_{k}(M)$ for the space of integral
$k$-currents in $M$
and write ${\cal D}^{\mathit{int}}...

**2**

votes

**0**answers

63 views

### Currents with mean curvature

so let $T=\tau(M,\theta,\xi)$ a rectifiable current in $U\subset\mathbb{R}^{n+k}$, then $T$ has mean curvature vector $H$ if for every $X\in C^1_c(U\setminus\partial T,\mathbb{R}^{n+k})$ the variation ...

**0**

votes

**1**answer

37 views

### Existence of stationary tangent cones

My question refers to Leon Simon's Book: Geometric Measure Theory, Chapter 42
So let $V\in G_n(U)$ a general Varifold and $\|\delta V\|$ the Total Variation measure. If the density $\theta^n(\mu_V,x)&...

**0**

votes

**0**answers

40 views

### Constancy theorem for integral currents

Please I need a reference where I can find a proof of the constancy theorem for integral currents which says : Let $A$ be a $C^1$ submanifold of $\mathbb{R}^n$ and $T$ an $m$-integral current such ...

**2**

votes

**0**answers

44 views

### decomposition of codim 1 currents

Let $T$ be an codimension one rectifiable mass-minimizing current. If $\partial T\llcorner B_R(a)=0$, then it is possible to write $T$ as an infinite sum of oriented boundaries: $T\llcorner B_R(a)=\...

**8**

votes

**1**answer

152 views

### Characterizing surface area

(This question is a variant of an unanswered question at math.stackexchange.)
The Definition section of Wikipedia's article on surface area currently starts as follows:
While the areas of many ...

**0**

votes

**0**answers

64 views

### Integral of gradient between level sets of Lipschitz functions

Start with a compact metric measure space $(X,d)$, with a doubling measure $\mu$ and a local regular Dirichlet form $\mathcal E$ that supports a Poincare inequality. $d$ can be taken to be the ...

**1**

vote

**0**answers

29 views

### Pointwise convergence of a sequence of approximate limits of BV functions

So, let $\Omega\subset \mathbb R^2$ bounded and consider a sequence of functions $\{u_k\}_{k\in\mathbb N}\subset BV(\Omega)$ and $u\in BV(\Omega)$ such that $u_k\rightarrow u$ weakly* in $BV(\Omega)$. ...

**0**

votes

**0**answers

78 views

### Hausdorff dimension and Hausdorff measure

Let us consider a curve $E\subset \mathbb{R}^2$ which is a $(\delta,R)$- Reifenberg flat domain and suppose also that it holds the following estimate on the Hausdorff dimension:$\mbox{ dim}_{\mathcal{...

**1**

vote

**2**answers

173 views

### Is there a name for this metric on a Borel sets

Consider a finite measure space $(X,\Sigma,\mu)$.
Consider the function $d:\Sigma \times \Sigma \to [0,1]$ given by
$$d(\sigma_1,\sigma_2) = \mu \left\{ (\sigma_1^c \cap \sigma_2) \cup (\sigma_1 \cap \...

**3**

votes

**0**answers

149 views

### Product Fractals

Here is a theorem found in the Falconer's book on fractal geometry:
Theorem: For any sets $E\subset \mathbb{R}^n$ and $F\subset \mathbb{R}^m$
$$
\dim_HF+\dim_HE\leq \dim_H(E\times F)\leq \dim_HE+\...

**0**

votes

**1**answer

68 views

### Approximating characteristic functions by cutting the real axis into smaller and smaller pieces

Let $\Lambda_r^*=\frac{1}{2\pi r} \mathbb{Z} \subset\mathbb{R} (r>0)$, let $E\subset\mathbb{R}$ be a Lebesgue measurable set with finite measure $|E|$, define $J_r=(-\frac{1}{4\pi r}, \frac{1}{4\pi ...

**2**

votes

**1**answer

114 views

### Almgren's mimeographed lectures notes on varifolds

I am trying to get some insights for the combinatorial argument of Pitts (in his PhD thesis 'Existence and regularity of minimal surfaces in Riemannian manifolds', Princeton University Press, 1981) to ...

**3**

votes

**0**answers

189 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 ...

**1**

vote

**0**answers

66 views

### A $\mathcal{C}^1$ domain and Hausdorff dimension estimate

Let us consider an open connected domain $E\subset \mathbb{R}^N$ and fix a point $x_0\in\partial E$ on its boundary.
Suppose now, that there exists $R>0$ such that the set $\partial E \cap B_{R}(...

**0**

votes

**0**answers

38 views

### A $\mathcal{C}^1$ differentiable domain is $F_\sigma$?

Let us consider a domain $E\subset \mathbb{R}^N$ and fix a point $x_0\in\partial E$ on its boundary.
Suppose now, that for every $R>0$ the set $\partial E \cap B_{R}(x_0)$ is $\mathcal{C}^1$, i.e. ...

**3**

votes

**0**answers

81 views

### Calculating the length of curve using dyadic partition [closed]

Let $\gamma:[0,1] \rightarrow \mathbb{R}^n$ be a continuous function. The length of $\gamma$ is usually defined as
$$\sup_{0 = t_1 < t_2 < \cdots < t_n = 1} \sum_{i=1}^{n-1} d_{\mathbb{R}^n}...

**0**

votes

**0**answers

68 views

### Can we represent a $N-1$ rectifiable set locally as a graph, with some price?

This is a follow up discussion for this post.
Let me copy part of the definition here.
Let $S\subset \mathbb R^N$ be a $N-1$ rectifiable set with $\mathcal H^{N-1}(S)<\infty$.
My (updated) ...

**2**

votes

**1**answer

108 views

### Lower bounds from Fourier dimension?

According to Mattila, Geometry of sets and measures in Euclidean spaces, p. 168, the Fourier dimension $\text{dim}_F(A)$ of $A\subseteq \mathbb R^n$ is the unique number in $[0,n]$ such that for any $...

**0**

votes

**1**answer

95 views

### The partition of $N-1$ rectifiable set

The updated version can be found here.
Let $S\subset \mathbb R^N$ be a $N-1$ rectifiable set with $\mathcal H^{N-1}(S)<\infty$.
My question, for each $x_0\in S$, would it be possible to choose a ...

**1**

vote

**0**answers

47 views

### Volume of intersection of a convex polytope with general affine space

This question generalizes (this question) on the same site.
Let $\Delta^{n}$ denote the $n$-dimensional simplex in $n$ dimensions.
That is, $\Delta^{n}$ is the convex closure of the origin
and the $n$...

**1**

vote

**0**answers

106 views

### Laplace method with “bad” zero set

It is well-known that if $\phi:\mathbb{R}^n \longrightarrow \mathbb{R}$ is a function with $\phi(0) = 0$, $\phi(x)>0$ if $x \neq 0$ and $D^2\phi(0) \geq 0$, then the integral
$$\int_{\mathbb{R}^n} ...

**5**

votes

**1**answer

224 views

### Convergence in the proof of Crofton's Formula

Let $\mathcal{L}$ be the set of oriented lines in $\mathbb{R}^2$ and let $\mu$ be the Kinematic Measure on $\mathcal{L}$; up to scaling, $\mu$ comes from the unique (up to scaling) volume form on $\...

**7**

votes

**1**answer

144 views

### Measure of chords from a cantor set

The following problem is inspired by a problem in Pugh's Mathematical Analysis book. (Chapter 2 Problem 42).
In the problem he asks one to consider the standard Cantor set on the unit interval, and ...

**14**

votes

**0**answers

115 views

### A kaleidoscopic coloring of the plane

Is there a partition $\mathbb R^2=A\sqcup B$ of the Euclidean plane into two Lebesgue measurable sets such that for any disk $D$ of the unit radius we get $\lambda(A\cap D)=\lambda(B\cap D)=\frac12\...

**4**

votes

**1**answer

134 views

### normal form of currents?

(this question did not get any answers on math.SE, so I am reposting it here)
Let $M$ be an $n$-dimensional manifold. Then the space of currents $\mathcal D^k(M)$ of degree $k$ on $M$ is the space ...

**6**

votes

**1**answer

141 views

### Existence of a measurable map between metric spaces

Let $X$ and $Y$ be separable complete metric spaces (if necessary, they may be assumed to be compact). Let $R\subset X\times Y$ be a closed subset such that the projection of $R$ to $X$ is onto.
Is ...

**3**

votes

**1**answer

98 views

### Reference request for a result regarding density of induced probability measure under a submersion

Let $\pi: M \to N$ be a smooth submersion from a bounded open subset of $\mathbb{R}^m$ onto $ N \subset \mathbb{R}^n$, $m \geq n$. Further, let $M$ be given a probability measure $\mu$. Then the map ...

**3**

votes

**1**answer

112 views

### Riemannian Measures, Densities and Radon–Nikodym Theorem

If $M$ is a smooth manifold and $\mu$ is a $1$-density thereon then we may define a Borel measure (on Borel sets $A$) on $M$ as:
\begin{equation}
\nu(A) = \int_M I_A \mu.
\end{equation}
My question ...

**6**

votes

**0**answers

94 views

### Convexity of Isoperimetric Domains

I am interested in what is known about the convexity of isoperimetric domains in compact Cartan-Hadamard manifolds (Riemannian manifolds that are complete and simply-connected and have non-positive ...

**1**

vote

**1**answer

31 views

### Multiplicity of a subcovering in spaces of given Hausdorff dimension

Let $X$ be a locally compact metric space of integer Hausdorff dimension $n$. Let $K\subset X$ be a compact subset. Let $\{B_i\}_i$ be a finite family of balls covering $K$. One may assume that all ...

**2**

votes

**0**answers

85 views

### $C^1$ Sard related question

Let $X$ be a $k+1$ rectifiable set with finite $k+1$ Hausdorff measure in $\mathbb{R}^{n+1}$ and set $Z=\{x\in X \mid e_{n+1}\perp T_xX \}$, where $T_xX$ is the approximate tangent and $e_{n+1}$ is ...

**3**

votes

**0**answers

211 views

### Area defined with $\pm$ closedness

Denote $B_n\subset\Bbb R^n$ to be unit ball at origin.
Denote $S\subset B_n$ to region of type $\mathsf I$ if it satisfies
$$s\in S\iff\forall t\in S, s+t\in S\mbox{ or }s-t\in S$$
I am convinced $\...

**1**

vote

**1**answer

72 views

### Image of a Jordan compact set under a degenerate map

This is crossposted from MSE, I hope this is suitable here, since there is no reaction there. I need this lemma for teaching, and I would appreciate any help.
Briefly:
Is the image of a Jordan ...