**0**

votes

**1**answer

27 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 ...

**0**

votes

**0**answers

30 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

39 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 ...

**8**

votes

**1**answer

137 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

59 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

25 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

65 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{ ...

**1**

vote

**2**answers

171 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

140 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 ...

**0**

votes

**1**answer

61 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

96 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

156 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

62 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 ...

**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

78 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} ...

**0**

votes

**0**answers

61 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

99 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

87 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

45 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 ...

**1**

vote

**0**answers

103 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

181 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

140 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

102 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 ...

**4**

votes

**1**answer

129 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

136 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

95 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

107 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

92 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

79 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

208 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

70 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 ...

**5**

votes

**1**answer

266 views

### Transcendental distance sets

Define a set $S \subset \mathbb{R}^d$ as a
transcendental distance set if the distance between any pair of
distinct points of $S$ is transcendental.
For example, $S = \{ k \, \pi \;\mid\; k=1,2,\ldots ...

**3**

votes

**2**answers

250 views

### What can we say about the boundary of the level set of a Sobolev function?

I'm a beginner of the area of free boundary problem. Let me first give some background:
$\Omega \subset \mathbb{R}^n$ is an open connected set, and locally $\partial \Omega$ is a Lipschitz graph.
...

**0**

votes

**0**answers

158 views

### What does the following space look like?

Pick fixed $a=(a_1,a_2,\dots,a_d)\in\{\pm1\}^d$.
Consider map $F_a:\underbrace{\Bbb R^n\times\dots\times \Bbb R^n}_d\rightarrow\Bbb R^n$ given by $F(x_1,\dots,x_d)=\sum_{i=1}^da_ix_i$.
Denote ...

**2**

votes

**0**answers

82 views

### What dimension bound is known on the singular set of a linear combination of eigenfunctions of Laplacian?

Let $(M,g)$ be a smooth, closed Riemannian manifold and suppose that $\phi_1,\dots,\phi_m$ are eigenfunctions of the Laplacian on $M$. Write $f = \phi_1 + \dots + \phi_m$.
How big can the set ...

**5**

votes

**0**answers

68 views

### Measure-minimizing simplex with fixed inradius

Let $\Delta^n$ be an $n$-simplex in $\mathbb{R}^n$. Let $V$ be the volume and $r$ the inradius (radius of the inscribed sphere) of $\Delta^n$. There is a well-known result that
$$
V \geq ...

**0**

votes

**0**answers

75 views

### Distortion of the Hausdorff dimension of sums of Cantor sets under local scaling

The following question deals with possible distortion of the Hausdorff dimension of sums of Cantor sets as one "zooms in" on the sum around any given point.
Let us assume that $C_1$ and $C_2$ are two ...

**9**

votes

**2**answers

215 views

### Continuity of length and area

Let $C_n$ be a sequence of rectifiable simple closed curves in $\mathbb{R}^2$ that converge to a rectifiable simple closed curve $D$ in the Hausdorff topology. It is easy to construct examples where
...

**2**

votes

**1**answer

60 views

### Convex hulls have longer boundaries

Let $C$ be a rectifiable simple closed curve in $\mathbb{R}^2$ and let $D$ be the boundary of the convex hull of the region bounded by $C$. What is the most efficient way to prove that $D$ is ...

**2**

votes

**3**answers

263 views

### When Banach indicatrix is measurable?

Let $f:X\to Y$ is a measurable function. Banach indicatrix
$$
N(y,f) = \#\{x\in X \mid f(x) = y\}
$$
is the number of the pre-images of $y$ under $f$. If there are infinitely many pre-images then ...

**12**

votes

**1**answer

252 views

### Best Hölder exponents of surjective maps from the unit square to the unit cube

The Peano's square-filling curve $p:I\to I^2$ turn's out to be Hölder continuous with exponent $1/2$ on the unit interval $I$ (a quick way to see it, is to note that $p$ is a fixed point of a ...

**1**

vote

**1**answer

165 views

### Singularities in minimal surfaces [closed]

There is a theorem by Jean Taylor that says that an almost minimal set in $\mathbb{R}^3$ can be locally parametrize by the only three possible minimal cones in $\mathbb{R}^3$, the plane, an $Y$ times ...

**0**

votes

**1**answer

121 views

### Probability Content of a random ball in R^n

As a follow up to this question, concerning this paper:
Given random variables $X_1,\ldots,X_N,X_q:\Omega\rightarrow\mathbb{R}^d$, where $X_1,\ldots,X_N$ are independent and identical distributed. ...

**4**

votes

**1**answer

221 views

### Hausdorff measure of the graph

Is there any example of a real valued function on the real line whose domain has Lebesgue measure zero but the graph (in the plane) has positive one dimensional Hausdorff measure?
Of course if such ...

**7**

votes

**3**answers

348 views

### How can dimension depend on the point?

Let $M$ be a metric space.
For any subset $A\subset M$ let $\dim(A)$ denote its Hausdorff dimension.
For $x\in M$, define the dimension of $M$ at $x$ by $\dim(x)=\lim_{r\to0}\dim(B(x,r))$; this limit ...

**2**

votes

**2**answers

102 views

### understanding the average height of a unit hyper-semisphere

According to my calculations, the average height of the unit hyper-hemisphere in $\mathbb{R}^n$ is given by the formula
$$h_n:=\frac{S_{n-2}}{V_{n-2}} \int_{r=0}^1 r^{n-2} \sqrt{1-r^2} dr$$
where ...

**2**

votes

**1**answer

130 views

### How often can a single length occur as a boundary distance?

Given a bounded domain $\Omega\subset\mathbb R^n$ ($n\geq2$), how often can a single real number $r>0$ appear as a distance of two points on $\partial\Omega$?
We can make any assumptions about the ...

**12**

votes

**2**answers

615 views

### What is the Hausdorff dimension of this fractal?

Let $\sum_{i=h}^\infty d_i/b^i $ be the base $b$ representation of $x \geq 0,$ where $b>1$ and the $d_i$ are uniquely determined by the greedy algorithm. For fixed $c>1,$ let $f(x)= ...

**3**

votes

**1**answer

71 views

### Subsets of sets of positive Hausdorff dimension with controlled upper Minkowski dimension

Call a Borel set $A \subseteq [0,1]$ good if $$0 < \dim(A) \leq \overline{\dim_\text{M}}(A) < 2 \dim(A),$$ where $\dim(A)$ is the Hausdorff dimension of $A$ and $\overline{\dim_\text{M}}(A)$ is ...