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Questions tagged [geometric-measure-theory]

The study of the geometric structure of measures, as well as the study of geometry from a measure-theoretic viewpoint, geometric measure theory has applications in partial differential equations, harmonic analysis, differential and Riemannian geometry, as well as calculus of variations. Statements such as the isoperimetric inequality and the coarea formula, and subjects such as the Plateau problem belong under this tag.

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Extending the Homotopy formula of Federer 4.1.9 to Riemannian Manifolds

Is it possible to extend the Homotopy Formula expressed for classical currents in open sets of $\mathbb{R}^n$ to currents in a Riemannian manifold? (see Federer's book Geometric Measure Theory 4.1.9) ...
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Box counting dimension of the graph of the Cantor function

Consider the Cantor staircase function. The Hausdorff dimension of its graph is $1$. What is the box-counting dimension of its graph? A more general question is on MathOverflow.
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15 views

Set of positive reach and Minkowski sum

Let us denote by $A\subset \mathbb{R}^2$ a set with positive reach and suppose $\rm{reach}(A) \ge r$. It is true then that the Minkowski sum $A\oplus B_r$ has regular boundary (say piecewise Lipschitz)...
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1answer
18 views

Hausdorff measure independent of metric

I have seen in several papers the claim that for a compact Riemannian manifold the Hausdorff measure will be independent of the Riemannian metric chosen. Could someone explain what this means ...
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1answer
120 views

Show that the probability of this sequence of random variables being a root of this function tends to $0$

Let $f\in C^3(\mathbb R)$ with $f>0$, $$\int f(x)\:{\rm d}x=1$$ and such that $(\ln f)'$ is Lipschitz continuous, $$p_n(x):=\prod_{i=1}^nf(x_i)\;\;\;\text{for }x\in\mathbb R^n$$ and $$h_n^{(x)}(z):...
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1answer
35 views

When $\mathbb E\int_0^T H_s^2ds=\int_0^T \mathbb E[H_s^2]ds$ is not true?

Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space and $(\mathcal F_t)_t$ a filtration. In all example I can see, we always have that $$\mathbb E\int_0^T H_s^2ds=\int_0^T \mathbb E[H_s^2]ds,$$ ...
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The Hausdorff dimension of the zero set of a real analytic function

Let $n>1$, and let $f:\mathbb{R}^n \to \mathbb{R}$ be a real-analytic function which is not identically zero. Does $\dim_{\mathcal H}(f^{-1}(0)) \le n-1$? here $\dim_{\mathcal H}$ refers to the ...
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Are the derivatives of the orthogonal polar factor locally integrable?

Let $\mathbb{D}^2$ be the closed unit disk, and let $f:\mathbb{D}^2 \to \mathbb{R}^2$ be a real-analytic map, satisfying $\det df>0$ everywhere except on a set of Hausdorff dimension not greater ...
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33 views

Prove that fractal sets do not have finite perimeter

How does one prove that fractal sets like the Koch snowflake, Mandelbrot set, Sierpinski triangle do not have finite perimeter in the sense of Caccioppoli? That is, how does one prove that their ...
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Functions of bounded variation and Mandelbrot set

Is the characteristic function of the Mandelbrot set a function of bounded variation?
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13 views

Prove that the characteristic function of Koch snowflake is not BV

We say that a measurable set is of finite perimeter if its characteristic function is of bounded variation. How does one prove that the total variation of the characteristic function of Koch ...
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1answer
41 views

Is $\mathrm{Cantor \, set} \times [0,1]$ self-similar?

Consider the Cantor set $\times$ the interval $[0,1]$, i.e. Cantor sets put "one next to the other" as to "cover" the quare $[0,1]\times [0,1]$ is this set self-similar, i.e. the attractor of some ...
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Heuristically, how can we visualize a tangent measure?

Heuristically, how can we visualize a tangent measure? Wikipedia makes an example in the case of the Hausdorff measure on the circle in $\mathbb R^2$, but I'm interested in visual examples in more ...
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1answer
41 views

Does the Lebesgue measure on the segment $y=x$ represent this distribution?

Set $\Omega=(-1,1)^2$. Consider the following measure on $\Omega$: $\mu(A)=m(A \cap L)$, where $L=\{ (x,x) \, | \, -1 < x < 1\}$ (the segment of the line $y=x$ in $\Omega$) , and $m$ is the ...
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31 views

A measure theoretic Lipschitz condition

Let $f$ be a measurable function satisfying following condition: for every $\epsilon$, we have \begin{equation*} \limsup_{\delta \to 0} \bigg\{ \frac{1}{\delta^N} \mathcal L^{2N} \Big( \Big\{ (x,y) \...
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1answer
85 views

Examples of BV functions $u:\mathbb{R}^2 \to \mathbb{R}^2$ with singular derivative

What are examples of two BV functions $u:\mathbb{R}^2 \to \mathbb{R}^2$ with singular derivative? More precisely, I'd like to see an example of a function $$u_1 \in BV(\mathbb R^2; \mathbb R^2)$$ ...
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1answer
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Does the domain where a harmonic map is conformal has Hausdorff dimension smaller than one?

Let $\mathbb{D}^2$ be the closed unit disk, and let $f:\mathbb{D}^2 \to \mathbb{R}^2$ be a harmonic map, satisfying $\det df \neq 0$ almost everywhere on $\mathbb{D}^2$. Suppose also that $$U= \{ p \...
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Definition of $\ell$-dimensional points [Geometric Measure Theory]

Reference: Cheeger, J., Colding, T., _On the structure of spaces with Ricci curvature bounded below., J. Differential Geometry, 45 (1997) 406--480. I have a question regarding the following ...
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1answer
44 views

lower semi-continuous map to a probability measure

Let $\mu$ be a probability measure and $R(p)$ an interval with the property that both $\inf(R(p)):=\underline{r}(p)$ and $\sup(R(p)):=\bar{r}(p)$ are continuous and bounded functions of $p\in [0,1]$, ...
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Locally bounded measure maximization

I am facing the following problem which I can not find a solution to, after several days of work : First, define the following : $\mu$ is locally bounded by $\omega$ if, for any subset $S$ of $\...
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How did Steiner prove his famous formula?

In convex integral geometry and geometric measure theory, Steiner's formula is the name of the following elegant result: Let $B_n$ be the unit ball in $\mathbf R^n$. If $S$ is a nonempty bounded ...
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26 views

Proving the Ham-Sandwich Theorem

A version of the "Ham-Sandwich theorem" states that if we have $n$ finite Borel measures $\nu_1 , \dots, \nu_n$ on $\mathbb{R}^n$ which assign zero measure to hyperplanes then there exists a halfspace ...
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1answer
49 views

Does there exist a compactly supported integrable function with infinite Coulomb energy?

The title of the question pretty much says it all. I am looking for a function $f\in L^1(\Omega)$, where $\Omega \subset \mathbb{R}^3$ is a bounded domain, such that $$ E[f] = \iint\limits_{\Omega\...
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1answer
30 views

Wasserstein 1-distance of push-forward measures

Suppose you are given two metric spaces $(X,d_{X})$ and $(Y,d_{Y})$ and a map $f:X \to Y$. Furthermore take two measures $\mu , \nu$ in $P_{1}(X)$ the Wasserstein 1-space over X. Let $\gamma \in \...
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Perimeter of level sets of a smooth function

I've a simple question concerning the perimeter of level sets of a smooth function. Let $f:\Omega \to \mathbb{R}$ be a smooth function defined on a bounded domain of $\mathbb{R}^n$. We set $A_s:=\{f&...
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1answer
50 views

Measures on infinite product of $[0,1]$?

What are the known measures defined on some sigma algebra on infinite product space of $[0,1]$ ? Is there any measure which is compatible with usual metric space structure of infinite product of $[0,...
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51 views

Lebesgue Measure in $\Bbb{R}^k$ is invariant under isometries

I'm studying Lebesgue Measure. I have a problem on proving that Lebesgue Measure in $\Bbb{R}^k$ is invariant under isometries Here is my work so far. Let $T$ $\mathbb{R}^k \to \mathbb{R}^k$ is an ...
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1answer
65 views

Define a measure in $\Bbb{R^k}$.

I'm studying Measure Theory. After reading my teacher's lecture notes, it is not clear for me the statement: "In $\Bbb{R}^k$, It's impossible to define a $\sigma$ finite measure the satisfies all the ...
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29 views

Intuition for definition of Radon Measure

According to the definition of the book "Measure Theory and fine properties of functions." -Evans Gariapery a measure u is said of Radon if they are worth the properties described in the attached ...
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1answer
41 views

Definition of $C^{1,\gamma}$-hypersurface

I'm studying by myself Geometric Measure Theory by the book "Sets of Finite Perimeter and Geometric Variational Problems An Introduction to Geometric Measure Theory" written by Francesco Maggi and he ...
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1answer
51 views

Nested family of sets with given Hausdorff dimension

Just before (namely here Existence of a set with given Hausdorff dimension) I asked whether one can find for any real number $\alpha>0$ a set $A_\alpha$ such that $A_\alpha$ has Hausdorff dimension ...
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1answer
36 views

Charges, Currents and Distributions: Terminology clarification.

I am trying to learn some measure theory by myself these days and, while reading some real analysis books I realize that some of the terminology is somehow related to electricity. For examples ...
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1answer
62 views

Maximizing the value of an integral

Let $f \colon \mathbb R^N \to \mathbb R$ be a measurable, bounded function. Let $$ \mathcal A := \left\{ g \colon \mathbb R \to [0,+\infty): g \text{ is measurable and} \int_\mathbb R g =1\right\}. $$...
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1answer
31 views

Can the rank of the differential of a Lipschitz map decrease in a small neighbourhood?

Is there an example for a Lipschitz map $f:\mathbb{R}^n\to\mathbb{R}^m$ which is differentiable at $x_o$, with $\operatorname{rank} Df(x_o)=k$, such that there is no open neighbourhood $U$ of $x_0$ ...
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1answer
46 views

Packing Dimension as a Countable Union of Minkowski Dimension Sets

Is it true that if $X$ has packing dimension $\alpha$, then we can write $X$ as the countable union of sets $X_i$, where $X_i$ has Minkowski dimension $\alpha$. If not, which notion of dimension is ...
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1answer
25 views

What is the product of two Haar distributed unitary matrices?

I guess a product of two Haar distributed unitary matrices is also a Haar distributed unitary matrix. Is there a proof?
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1answer
63 views

How did we resolve the Banach-Taraski paradox?

I see that when Banach-Taraski paradox emerged we solved this problem by stating that not every subset is measurable so we restrict ourselves to nice sets which are measurable. But How? I'm confused a ...
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24 views

Definition of Borel vector field

My doubt is what is the definition of Borel vector field? I looked for this definition on some books of geometric measure theory and on the internet, but I didn't found. The motivation for my question ...
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2answers
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Is there a generalization of the nested interval theorem in $\mathbb{R}^n$?

I'm familiar with the nested interval theorem on the real line. But is there a generalization of such a theorem in literature?
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1answer
55 views

Prove the geometric “Pigeonhole Principle”

This question is part of an introductory combinatorics class, so I don't know what measure theory is, but the question was stated as follows: Suppose $A_1, A_2,... A_k, B$ are sets which contain a '...
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1answer
27 views

Integrating the composition of a Heaviside function with a smooth function

I am trying to find how to compute an integral of the form: $\int_{R^n}{\Theta(g(x))f(x)\,dx}$, where $\Theta$ is the Heaviside function, $g(x)$ is a smooth function (a result for more general $g$ is ...
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0answers
20 views

Natural way of thinking of the definitions of the rectifiable sets and purely unrectifiable sets

I'm selfstudy Geometric Measure Theory by Frank Morgan's book and he define rectifiable sets as follows A set $E \subset \mathbb{R}^n$ is called $(\mathscr{H}^m,m)$ rectifiable if $\mathscr{H}^m(E) ...
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Integer multiplicity current without boundary is boundary of another current

Let us say we have an integer multiplicity current $T\in \mathcal{D}_n(\Omega)$, $\Omega\subset \mathbb{R}^m$, $n+1\leq m$ and $\partial T =0$. Do we always find another current $R\in \mathcal{D}_{n+1}...
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Is it true that $\mathcal H^{n-1}(\partial (A \cap B))=\mathcal H^{n-1}((\partial A) \cap B) + \mathcal H^{n-1}( A\cap ( \partial B))$?

Let $A$ and $B$ subsets of $\mathbb R ^n$, $B=B(x,r)$ an open ball, and denote the $(n-1)$-dimensional Hausdorff measure in $\mathbb R ^n$ by $\mathcal H^{n-1}$. Also assume that $\mathcal H^{n-1} (\...
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3answers
207 views

Compute area of a sphere through a Dirac delta

I've been having issues with integrating with a Dirac delta. To compute the area of a sphere centered at $(0,0,0)$ it seems to work just fine: $$\int_{0}^{2\pi}{\int_{0}^{\pi}{\int_{0}^{\infty}{\...
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0answers
54 views

What motivates the use of the co-area formula? What is co-area?

The motivation for the area formula is very clear. Given a set A in $\mathbf{R^n}$, and a Lipschitz function f $: \mathbf{R^n} \rightarrow \mathbf{R^{N}}$, what is the measure of the set f(A) in $R^{N}...
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2answers
83 views

A counter-example for integration by parts when there are “small” singularities

I am looking for a "counter-example" to integration by parts of the following type: $\Omega \subseteq \mathbb R^n$ is an open, bounded, connected domain with smooth boundary. $u,v:\bar \Omega \to \...
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24 views

Lipschitz image $\mathbf{R} \to \mathbf{R}^2$ has measure zero?

I know that of a proof of the following fact using Hausdorff measure, but is there a more elementary way to do it? Suppose $f: \mathbf{R} \to \mathbf{R}^2$ is Lipschitz. Then $\mathcal{L}^2(f(\...
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1answer
171 views

Is this function continuous/smooth?

Suppose $A \subset \mathbb{T}_2$ is a measurable set, where $\mathbb{T}_2$ is the torus group. For a fixed $n$, define the function $f_A:\mathbb{T}_2^n \rightarrow \mathbb{R}$ as $$f_A(x_1, ..., ...
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91 views

Best topics to study in order to research geometric measure theory

What are the best topics (other than GMT itself) to learn in order to pursue research in the field of geometric measure theory?