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Questions about geometric properties of sets using measure theoretic techniques; rectifiability of sets and measures, currents, Plateau problem, isoperimetric inequality and related topics.

5
votes
1answer
96 views

The Hausdorff dimension of the union of singular orbits and exceptional orbits

Suppose we have a compact connected Lie group $G$ acting as isometries on a compact manifold $M^n.$ Then is it necessarily true that the Hausdorff dimension of the union of singular and exceptional ...
6
votes
0answers
87 views

Does an 8 dimensional compact Riemannian manifold contain an embedded minimal hypersurface?

It is well know that if $(M^{n+1},g)$ is a compact Riemannian manifold and $n \leq 6$ then there exists a smooth, embedded minimal hypersurface $\Sigma^n$ in $M$ (infinitely many such $\Sigma$, even) ...
3
votes
0answers
55 views

Integration on a family of differential forms

Let $X$ be a smooth manifold, and denote by $\Omega^*(X)$ the set of all smooth differential forms on $X$. Assume we have a family of differential forms $\omega_t \in \Omega^*(X)$, $t\in E$, ...
1
vote
0answers
74 views

Egorov's and Lusin's Theorem in the space with infinite measure

Both the fundamental Egorov's and Lusin's Theorem in measure theory are given on any measurable space $X$ whose measure is finite. On the measurable space whose measure is infinite, does there ...
2
votes
0answers
46 views

Volume of critical points decreases under symmetric decreasing rearrangement

In the lecture note http://www.math.utoronto.ca/almut/rearrange.pdf, it was stated that the volume of the set of critical points decreases under symmetric decreasing rearrangement. It seems so obvious ...
6
votes
3answers
200 views

about the Hausdorff dimension of Removable singularities of PDE

There are some interesting phenomenons about removable singularities (or extension problems). In the theory of functions of several complex variables, we know the classical Hartogs theorem: Let f ...
2
votes
0answers
110 views

Lebesgue density theorem for “doubling uniformly covering collections of subsets”

I am looking for a version of Lebesgue density theorem that works when restricting to "good" collections of balls with respect to (not necessarily doubling) metric measure spaces. Specifically Let $(...
1
vote
0answers
47 views

about the compactness of minimal surfaces

If a Caccioppoli set $A$ is of minimal perimeter in every compact set $K$ contained in some open set, can we say that $A$ is of minimal perimeter in the open set? If not, please construct a ...
0
votes
0answers
143 views

On the proof of Modified Vitali Lemma

I see the following theorem in Lihe Wang's A geometric approach to the Calderon--Zygmund estimates (Modified Vitali) Let $0<\varepsilon<1$ and let $C\subset D\subset B_1$ be two measurable ...
2
votes
1answer
57 views

Existence of a Lipschitz map from a positive measure set to a ball

Question. If $A\subset \mathbb{R}^n$ is any set of positive Lebesgue $n$-measure, does there exists a Lipschitz map $f:A\to\mathbb{R}^n$ such that $f(A)$ is a ball with the same measure? In dimension ...
6
votes
1answer
116 views

Transportation-cost inequality for pushforward measure

Let $X=(X,d_X)$ and $Y=(X,d_Y)$ be metric spaces and $\varphi: X\rightarrow Y$ be an $L$-Lipschitz map, with $0 \le L < \infty$. Suppose $\mu$ is a probability measure on $X$ which satisfies ...
0
votes
0answers
93 views

Is there a precise relationship between ``Geometric Functional Analysis" and high-dimensional probability/information theory?

The 2009 course on GFA by Roman Vershynin (https://www.math.uci.edu/~rvershyn/papers/GFA-book.pdf) introduced the subject with this line on the course page, "...
7
votes
1answer
258 views

Axioms of length

Assume I want to define length of plane curves axiomatically. It seems to be reasonable to assume that The length of a unit segment is 1; Congruent curves have equal lengths; Length is additive with ...
2
votes
1answer
69 views

A property of Lipschitz domains

I have a question about a property of Lipschitz domain. Let $D \subset \mathbb{R}^d$ be a bounded domain (connected open subset ). $D$ is called a bounded Lipschitz domain if there exist positive ...
1
vote
1answer
107 views

Wasserstein interpolation between two probability measures on a metric space

Question 1 Given probability measures $\mu$ and $\nu$ on the same metric space $X=(X,d)$, and $\alpha \in [0, 1]$, is it always possible to find another probability measure $\lambda_\alpha$ on $X$ ...
0
votes
0answers
37 views

Reformulate Wasserstein constraint optimization on product space in terms of marginal

Let $X = (X,d_X)$ be a metric space and $Y$ be an abstract set (with at least two elements). Consider the metric on $X \times Y$ defined by $$d((x,y),(x',y')) = \begin{cases}d_X(x,x'),&\mbox{ if }...
1
vote
0answers
41 views

From Sudakov minoration principle to lowerbounds on Rademacher complexity

For a compact subset $S \subset \mathbb{R}^n$ (and an implicit metric $d$ on it) and $\epsilon >0$ lets define the following $2$ standard quantities, Let ${\cal P}(\epsilon,S,d)$ be the $\epsilon-...
9
votes
1answer
247 views

Current vs Varifold

I know the basic definitions concerning current and varifold, and they are generalization of submanifolds. What are their respective pros and cons? What are their crucial similarities and differences?
1
vote
0answers
35 views

Change of variables between quadrilaterals - Rayleigh quotient

A - Vertex at bottom left B - Vertex at bottom right K - Vertex at top left of blue quadrilateral C - vertex at top left of brown quadrilateral L - vertex at top right of blue quadrilateral F - ...
9
votes
1answer
253 views

Can a big set always look small?

For a set $C\subset \mathbb R^2$, define its visibility from a point $x$ as $vis_C(x)=\{\varphi\in \mathbb S^1\mid \exists t>0~~x+t*\varphi\in C\}$, where $\mathbb S^1$ denotes the unit circle. Say ...
3
votes
1answer
111 views

Nice representation of open sets in $\sigma$-algebras in certain circumstances

Let $(X,\tau)$ be a topological space. For a given topological base $\mathcal{E}$ for $\tau$, let us denote Bor$(\mathcal{E})$, by the smallest $\sigma$-algebra containing $\mathcal{E}$. Q. Assume ...
4
votes
1answer
83 views

Nice arrangement of open sets in $\sigma$-algebras

Let $X$ be a topological space and $\mathcal{E}$ be a topological base for $X$. Let us denote Bor$(\mathcal{E})$, by the smallest $\sigma$-algebra containing $\mathcal{E}$. Q. Let $O$ be an open ...
4
votes
1answer
97 views

Weak convergence of measures on dense sets

We are given a complete (separable) metric space $X$ and a dense subset $D\subset X$. Consider a sequence of continuous functions $f_n\colon X\to \mathbb R$ such that $$\int\limits_D f_n \, {\rm d}\mu\...
3
votes
1answer
104 views

Approximation on separable topological space with size $\mathfrak{c}$

Let $X$ be a separable topological space of size $\mathfrak{c}$. By a simple function $\phi:X\to X$, we mean a finite range valued measurable function. Q. Is it possible to find a sequence of ...
1
vote
0answers
125 views

Comparing two $\sigma$-algebras

Let $X$ be a set. We denote $P(X)$ by the family of all subsets of $X$. We also denote $P(X)\otimes_{\sigma}P(X)$ by the $\sigma$-algebra generated by $\{A\times B: A,B \subseteq X\}$. Q. For which ...
5
votes
0answers
99 views

Smoothing properties of convolutions of $P^1(\mathbb{R})$ by $SL(2,\mathbb{R})$

Consider the action of $SL_2(\mathbb R)$ on real projective space $P^1(\mathbb R)$; given $A \in SL_2(\mathbb R)$ and $\alpha \in P^1(\mathbb R)$ we write $A . \alpha \in P^1(\mathbb R)$ for this ...
8
votes
1answer
182 views

Is there a non-atomic finite positive measure in the plane, of which uncountably many projections have atoms?

I would like to know whether or not there exists a finite probability measure $\mu$ on $\mathbb R^2$ which has no atoms, but such that there exists an uncountable set $A\subset \mathbb S^1$, such that ...
5
votes
1answer
92 views

Second fundamental form blows up at minimal hypersurface singularity

I have seen the claim that "$A$ bounded on an area minimizing current implies no singular set" in a couple of papers by Lohkamp, but with no reference (see https://arxiv.org/abs/1805.02180 e.g.). ...
1
vote
0answers
61 views

Integral of the square of the areas of slices of a shape

Suppose $\omega$ is a bounded shape in $\Bbb{R}^3$ and that $\{z : (x,y,z) \in \omega \}=[0,T]$ (that is, the shape is exactly contained in the band $\{z \in [0,T]\}$. If we denote by $\omega_t = \{(x,...
1
vote
1answer
124 views

The product of two controlled operators is also a controlled operator

The following picture is lemma 4.23 in Lectures on Coarse Geometry by John Roe: I guess the $E_i$ in the centered formula is $X_i$. Does Roe mean that $X_j\cap \mathrm{Supp}(u)=\emptyset $ implies $\...
2
votes
2answers
89 views

What are some applications of Dilation Structures(idempotent right quasi-groups) from Emergent Algebra?

According to the following Journal Articles, there are these structures called Dilation Structures that are formalised in Emergent Algebras, examined in the case of metric spaces with dilations, and ...
5
votes
0answers
198 views

When are Lipschitz functions dense in continuous functions?

Let $X$ be a compact metric space, and let $Y$ be another metric space. I am looking for examples of, and especially references to, theorems that give conditions under which any continuous mapping $f:...
2
votes
0answers
60 views

Points on Sphere whose image, under symmetric positive definite matrix, is contained in cube

Let $\Sigma \in \mathbb{R}^{n \times n}$ be a symmetric, positive definite matrix and let $\mu_r$ denote surface measure on the sphere in $\mathbb{R}^n$ with radius $r$. Let $$ R = \{x \in \mathbb{R}^...
13
votes
3answers
442 views

Is the intersection of two Caccioppoli (i.e. finite perimeter) sets Caccioppoli?

Recall that we say that a bounded measurable set $S\subset\mathbb R^n$ is said to be Caccioppoli if the indicator function $1_S$ is BV, and we set $$ \operatorname{perim}(S)=\| \nabla 1_S\|_{TV} $$ ...
1
vote
0answers
61 views

Showing that $b$ is a inner point of $\mathcal{G}$ where $\mathcal{G}$ is a subset of $\mathbb{R}^{N+3}$ determined by $\mathcal{M}^{+}$

Let $(\Xi,\mathscr{E})$ be a measurable space, $(\mathbb{R_{+}},\mathfrak{B})$ other measurable space where $\mathfrak{B}$ a $\sigma$-algebra. We consider the measurable space $(\Xi\times\Xi\times\...
4
votes
2answers
123 views

Mean width and perimeter

Does anyone know a simple, elementary and self-contained proof of the fact that the mean width of a convex two-dimensional body equals its perimeter divided by $\pi$?
7
votes
0answers
189 views

Isoperimetric inequality and geometric measure theory

The following version of the isoperimetric inequality can be easily deduced from the Brunn-Minkowski inequality: Theorem. If $K\subset\mathbb{R}^n$ is compact, then $$ |K|^{\frac{n-1}{n}}\leq n^{-1}...
3
votes
1answer
94 views

Geometric mean of positive measures

Let me given with an obvious example. Let $\Omega\subset{\mathbb R}^n$ be an open domain. If $f,g\in L^1(\Omega)$ and $f,g\ge0$, then $\sqrt{fg}\,\in L^1(\Omega)$. Now let me replace the absolutely ...
6
votes
0answers
235 views

What’s the limit of a vector bundle?

In geometric measure theory, there’s an answer to the question “what’s the limit of a family of submanifolds”, namely there’s some kind of object called an integral current. In the geometric ...
1
vote
0answers
144 views

Compact sets of Hausdorff dimension zero

I have a question about Hausdorff dimension. Suppose S is a compact subset of $\mathbb{R}^n$ whose Hausdorff dimension is zero. Does it follow that S can be covered by a finite DISJOINT union of ...
3
votes
1answer
91 views

Relative volume increase of $\delta$-fattening of a connected set

The following question was asked very recently at Relative volume increase of δ-fattening of a compact set: Is the following inequality true for all non-empty, compact sets $A \subseteq \mathbb{R}^n$ ...
3
votes
1answer
120 views

Surface/Volume-Ratio of an $\epsilon$-extension of a compact subset $S \subset \mathbb R^n$

For a non-empty, compact set $S \subset \mathbb{R}^n$, the $\epsilon$-extension of $S$, $S_\epsilon$, is defined to be the set $$ S_\epsilon = \cup_{a \in A} B_{\epsilon}(a), $$ where $B_\epsilon(a)$ ...
5
votes
1answer
216 views

Relative volume increase of $\delta$-fattening of a compact set

For a non-empty, compact set $A \subseteq \mathbb{R}^n$, the $\delta$-fattening of $A$, $A_\delta$, is defined to be the set $$ A_\delta = \cup_{a \in A} B_{\delta}(a), $$ where $B_\delta(a)$ denotes ...
4
votes
0answers
48 views

Tangent distribution for particular non-doubling measure (GMC)

The radon measure $\mu$ on [0,1] called GMC (Gaussian multiplicative chaos) satisfies the following: $$(1/c)|I|^{a}\leq\mu(I)\leq c|I|^{b},$$ $$\sup_{x\in [0,1]}\frac{\mu(B_{2r}(x))}{\mu(B_{r}(x))^{1-...
2
votes
0answers
57 views

Are these two sets always coincide after a translation or scaling?

I get stuck with the following problem, which I think is related to sum-product estimate. Here is the problem. Problem Given two sets $A, B\subset \mathbb R^n$, and a sires of positive number $\...
1
vote
1answer
179 views

From measurable to quantitative estimates of a map in the coarea formula

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m $ be Lipschitz and $n \geq m$. A version of the coarea formula says: $$ \int_A g(x) J_m f(x) d \mathcal{L}^n (x) = \int_{\mathbb{R}^m } \int_{ A \cap f^{-...
10
votes
1answer
295 views

Structure of the Cantor part of the derivative of a BV function

It is well known that an integrable function $u \colon \mathbb R^d \to \mathbb R$ is said to be of bounded variation iff the distributional gradient $Du$ is (representable by) a finite Radon measure, ...
4
votes
1answer
253 views

Is there a measure on the sphere with positive Fourier transform?

Is it possible to have an even probability measure $\mu$ (that is $\mu(A)=\mu(-A)$ for any set $A\subset \mathbb{R}^d$) supported on the unit sphere $S^{d-1}$ such that its Fourier Transform $$ \...
1
vote
0answers
64 views

Measure of the boundary of the support of a certain function defined by an expectation

Suppose: $\mathcal{S} = \{ S \in \mathbb{R}^d \ | \ S_i > 0, \forall i = 1,...,d \} $ $R$ is a random vector (on some probability space, $\Omega$) such that, $R: \Omega \to \mathcal{S}$. $h : ...
1
vote
0answers
110 views

Approximation of Borel sets

Let $\nu$ be a finite Radon measure on $\mathbb{R}^2$ and denote the Lebesgue measure on $\mathbb{R}^2$ by $\mathcal{L}^2$. Assume that $\nu<<\mathcal{L}^2$. We denote the boundary of $A\subset\...