# Questions tagged [geometric-measure-theory]

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

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185 views

### Textbook recommendation request: Exercises to supplement Evans and Gariepy

While a great book about measure theory and real analysis in $\mathbb R^n$, the only downside is the lack of exercises. Can anyone provide a good book to supplement it with exercises? I plan to use it ...

**1**

vote

**1**answer

113 views

### Reference request: Functions of bounded variation in one real variable

Is there a good reference for facts and theorems about BV real valued functions? I’m looking for something with much more than say Stein and Shakarchi 3, or Evans and Gariepy. Thanks!

**0**

votes

**0**answers

48 views

### convergence of a fraction containing Lebesgue measures and perimeters

Sorry for this basic question posted on MO, I as well asked here https://math.stackexchange.com/questions/3063202/convergence-of-a-fraction-containing-lebesgue-measures-and-perimeters
Let $\epsilon&...

**6**

votes

**1**answer

129 views

### Anisotropic perimeter and regularity of anisotropic minimal surfaces

1. Introduction.
By-now classical results assert that minimal surfaces (in $\mathbb R^n$) are generically "smooth" out of a "small" set.
Question. What are the known regularity results for ...

**2**

votes

**1**answer

80 views

### Generalizing a notion of purely unrectifiable sets

The Besicovitch-Federer structure theorem enables us to make the following definition:
Suppose $k < n$, $E \subset \mathbf{R}^n$, and $0 < \mathcal{H}^k(E) < \infty$. We say $E$ is purely ...

**5**

votes

**2**answers

139 views

### Origin of term Ahlfors-David regular

Much of the literature on analysis in metric spaces makes use of an assumption called Ahlfors regularity or Ahlfors-David regularity. Let $q>0$. A metric space $(X,d)$ is Ahlfors(-David) $q$-...

**4**

votes

**1**answer

128 views

### Flat norm metrizes the weak* topology

I've come across the following statement in literature (without proof or reference) about the flat norm of currents
$$
F(T) = \sup \{ T(\omega) : \omega \in D^k(U), |\omega(x)| \leq 1, |d\omega(x)| \...

**2**

votes

**1**answer

104 views

### Difference quotient for functions of bounded variation

Let $u:\mathbb{R}^N \to \mathbb{R}^N$, $u \in BV(\mathbb{R}^N)$, be a function of bounded variation.
We have that the following holds
$$(\ast) \qquad \frac{1}{|B_r(0)|}\int_{B_r(0)} \frac{|u(x+z)-...

**1**

vote

**0**answers

107 views

### How to show that this function is continuous (Geometric Measure Theory)

I want to prove that the function $F: \mathbb{R}_+ \to \mathbb{R}$ defined by
$$F(t)=\int_{\{d=t\}} g \, d\mathcal{H}^{n-1}$$
is continuous if $g:\Omega \subset \mathbb{R}^n\to \mathbb{R}$ is ...

**2**

votes

**0**answers

86 views

### Is a maximal set of rectangles known for which Lebesgue’s Differentiation Theorem holds true?

Lebesgue's differentiation theorem states that if $x$ is a point in $\mathbb{R}^n$ and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a Lebesgue integrable function, then the limit of $\frac{\int_B f d\...

**2**

votes

**1**answer

223 views

### Meaning of Alberti rank-one theorem

Heuristically what does Alberti's rank-one theorem imply about the structure of a $\mathrm{BV}$ vector field $\boldsymbol{b}$?
Is it rigorously fair to say that the level lines of $\boldsymbol{b}$ ...

**5**

votes

**1**answer

111 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

**0**answers

95 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

**0**answers

57 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

**0**answers

114 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

**0**answers

47 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

**3**answers

220 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

**0**answers

114 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

**0**answers

50 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 ...

**2**

votes

**1**answer

59 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

**1**answer

119 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

**0**answers

98 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

**1**answer

262 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

**1**answer

79 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

**1**answer

120 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 $X$ ...

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votes

**0**answers

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

**0**answers

48 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**

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**1**answer

257 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**

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36 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

**1**answer

255 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

**1**answer

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

**1**answer

84 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

**1**answer

103 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**

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**1**answer

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 ...

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**0**answers

128 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 ...

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votes

**0**answers

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**

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**1**answer

184 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

**1**answer

138 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.).
...

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vote

**0**answers

63 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,...

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**1**answer

126 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 $\...

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votes

**2**answers

90 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

**0**answers

229 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

**0**answers

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

**3**answers

517 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
$$
\operatorname{perim}(S)=\| \nabla 1_S\|_{TV}
$$
...

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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\...

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votes

**2**answers

132 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**

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**0**answers

195 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

**1**answer

95 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

**0**answers

243 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**

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147 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 ...