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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 geometry, Riemannian geomerty, sub-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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Why is studying upper bounds for $|I_\delta(\mathcal P,\mathcal L)|$ useful?

A natural problem in incidence geometry is counting the number of incidences of points and lines. For example, if $\mathcal P$ is a collection of points in $\Bbb R^d$, and $\mathcal L$ is a collection ...
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Is the $\alpha$-Hausdorff content of a set of diameter $\delta$ equal to $\delta^\alpha$?

Fix $\alpha>0$. The $\alpha$ dimensional Hausdorff content of a bounded set $K$ is given by $$H^{\alpha}_{\infty}(K)=\inf\left\{\sum_{i=1}^\infty \text{diam}(B_i)^\alpha:K\subset\bigcup_{i=1}^\...
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The measure of what sets is uniquely determined by _finite_ additivity (and translation invariance and normalisation)?

I am very familiar with measure theory but am currently wondering about how far finitary methods can take you. Two aspects have to be differentiated: the unique determination and the calculation of ...
justanotherhumanbeing's user avatar
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If a convex set E is contained in a unit ball B, with vol(E)/vol(B)>1-\varepsilon, can we find r close to 1 such that E contains B(0,r)?

Suppose that $E$ is a convex set and $B$ is the unit ball in $R^n$. Let $\varepsilon$>0 be a constant small enough. If we have $\frac{vol(E)}{vol(B)}>1-\varepsilon$, can we find $r>0$ close ...
ccs's user avatar
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Collections of distinct subsets of an interval (existence)

Question: does there exist an uncountable family of distinct subsets of $[-1, 1]$, denoted by $(U_j)_{j \in [-1, 1]}$, with the following properties? Throughout, $\mu$ is just the Lebesgue measure. ...
Stepan Plyushkin's user avatar
3 votes
1 answer
108 views

Stokes theorem for currents on manifolds with corners

Let $M\subset\mathbb R^N$ be a compact oriented $n$-(sub)manifold with corners and $\omega$ be an $(n-1)$-form on it. The usual statement of Stokes theorem $$\int_M d\omega=\int_{\partial M}\omega$$ ...
Derivative's user avatar
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Integral of log-concave function on the boundaries of convex sets

Let $g$ be a convex function on $\mathbb{R}^d$ and $B_1 \subset B_2 \subset \mathbb{R}^d$ be two convex sets with smooth boundaries $\partial B_1, \partial B_2$. Then do we always have the following ...
HenryYRZ's user avatar
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How to prove that $\left(\frac{r_1}{r_2}\right)^n{\rm vol}(\Omega_{r_1})\leq{\rm vol}(\Omega_{r_2})\leq{\rm vol}(\Omega_{r_1})$ for $0<r_1\leq r_2$?

Let $\Omega\subset \mathbb{R}^n$ be a bounded connected domain. Introduce the notation $$ \Omega_r=\left\{x\in\Omega|d(x,\partial\Omega)<\frac{1}{r}\right\},\quad r>0, $$ where $d(x,\partial\...
Kimura Leo's user avatar
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Hausdorff measure and orthogonal projection

I was reading Federer's proof of Gustin's boxing inequality and during the proof there's a step which is unclear to me. Here's the context: $A,B\subseteq \mathbb{R}^{n}$ compacts sets such that $A\cup ...
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About the sigma algebra generated by the Hausdorff measure on $\mathbb R^n$

Let $\mathcal{H}^k$ be the $k-$dimensional Hausdorff measure on $\mathbb R^n$, with $k \in \{1, \ldots n\}$. By Carathéodory's theorem we know that there exists a sigma algebra $\mu(\mathcal{H}^k)$ of ...
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Bounds on the volume of the image of a cube, through bounds on area of cross sections

Let $f:[0,1]^4\to\mathbb{R}^4$ be a smooth map. Assume that for every “vertical” affine plane $F=\{(a,b)\}\times [0,1]^2$, Area$(f[F])< a$. Assume also that for every “horizontal” affine plane $E= [...
JustSomeGuy's user avatar
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Image of a $C^1$ manifold

Let $F: \mathbb{R}^n \to \mathbb{R}^{n+k}$ be a $C^1$ map with $n, l >0$. It's well know that $F(\mathbb{R}^n)$ is not necessarily a $C^1$ manifold of $\mathbb{R}^{n+l}$. Then what about the ...
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Question in the proof of the Riesz Representation theorem of non-negative functionals in geometric measure theory written by Leon Simon

The problem is from the proof of Theorem 1.5.12 in Leon Simon's book: Geometric Measure Theory Suppose $X$ is a locally compact Hausdorff space, $\mathcal{K}^{+}$ is the set of all non-negative ...
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What is the dual space of $C^1_0(\Omega)$ for open $\Omega \subset \mathbb{R}^n$?

What is the dual space of $C^1_0(\Omega)$ for open $\Omega \subset \mathbb{R}^n$, where $\Omega$ is an open, possibly unbounded set, and $C^1_0(\Omega)$ is equipped with the supremum norm $\|f\|:=\...
1Rock's user avatar
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Question in the proof of the Riesz Representation theorem (locally compact Hausdorff case)

I'm reading the proof of Riesz Representation theorem (1.5.14) in Leon Simon's book: Geometric Measure Theory Previously I've shown that $|L(fe_{j})|\leq \int_{X}|f|d\mu=\|f\|_{L^{1}(\mu)}$ for all $...
OneLamp's user avatar
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Can we show $I_s(\mu) \lesssim_{\mu} I_t(\mu)$ when $0 < s < t$? When is $\frac{I_s(\mu)}{I_t(\mu)} \le 1$?

Let $\mu$ be a Borel measure on $\Bbb R^n$, with compact support $\operatorname{spt} \mu$. I'd like to show that if $0 < s < t$, then $I_s(\mu) \lesssim I_t(\mu)$. In particular, I'd like to ...
stoic-santiago's user avatar
6 votes
2 answers
139 views

Reference for Surface area measure

Can someone please help me with some good references on Surface area measures for open, connected sets? What I meant by this is the following: Suppose $\Omega$ is a bounded open, connected set in $\...
Zack math's user avatar
1 vote
1 answer
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The proof of the Riesz representation thoerem for non-negative functionals

I was reading Introduction to Geometric Metric Theory by Leon Simon. Here's the full statement of the Riesz representation theorem for non-negative functionals. Suppose $X$ is a locally compact ...
OneLamp's user avatar
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Jacobian of seminorm on $\mathbb R^n$

As follows $\mathcal{H}^n$ denotes the $n-$dimensional Hausdorff measure and $\omega_n$ denotes the volume of the unit ball on $\mathbb R^n$. In the article Rectiable sets in metric and Banach spaces ...
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Definition of $L^2(S^{n-1})$

I'm puzzled with how we're supposed to define $L^2(S^{n-1})$ where $S^{n-1}$ is the unit sphere in $\mathbb{R}^n$. How do we even define the inner product there? the only way that comes to mind ...
hteica's user avatar
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Are these two function spaces identical?

Let the function space $A$ denote all functions $f : [0, 1) \to [0, 1)$ such that, for some set $Z$ of Lebesgue measure zero, the derivative $f'$ exists on $[0, 1) \setminus Z$ and $|f'| = 1$ there. ...
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Fourier transform as an integral on a surface

Let $v\in C_c^\infty(\mathbb{R}^n)$ with support $B\subset \mathbb{R}^n.$ Let $h$ be real valued smooth function on a neighborhood of $B$ and $\xi \in \mathbb{R}^n, t\in \mathbb{R}$ and consider the ...
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Equality involving an SBV (Special Bounded Variation) function and an $L^{\infty}$ function

The notations are mainly those of Functions of Bounded Variation and Free Discontinuity Problems by Ambrosio, Fusco and Pallara. Hi, In a problem I am considering, I have reached the following ...
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incidence coefficients in homological integration

In the theory of CW complexes, the incidence coefficient between a cell $\sigma$ of dimension $k$ and a cell $\tau$ of dimension $k - 1$ of a complex is defined as the topological degree of the map $\...
Daniel Shapero's user avatar
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Manhattan distance between point and line

What is the minimum manhattan distance between a point A, with coordinates (i,j) and a point B on line y=mx+B In other words: $$\operatorname{min}{[\operatorname{D_T}{(A,B)}]}$$ Where $B=(x,mx+b)$ ...
Yashvasin Hariharan's user avatar
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1 answer
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Understanding Falconer Example 4.2

Presently, I am reading Falconer's book on Geometric Measure theory. In example 4.2 he used mass distribution principle to calculate the lower bound of the Hausdorff dimension of Cantor set $C$. Now ...
Mayank's user avatar
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1 answer
271 views

Stokes' theorem in differential geometry vs measure theory [closed]

For some context, I'm an undergraduate student in mathematics and have attended an introductory measure theory lecture. The lecture seemed to loosely follow some chapters of Real Analysis (Folland, ...
sinopterus's user avatar
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Statistical invariants of Riemannian Manifolds

A cheap way of defining invariants of Riemannian manifolds? Let $(M, g)$ be a Riemannian Manifold with finite volume and $X=(X_1,\dots, X_n)\sim \mathscr U(M^n)$ a random sample with respect to the ...
Alex's user avatar
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Does this measure have a singular continuous component?

The Lebesgue decomposition theorem says that any $\sigma$-finite measure can be decomposed into the sum of: an absolutely continuous measure, a singular continuous measure, and a discrete measure. I'...
Ryan's user avatar
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Functional for the prescribed mean curvature

Let $F:M\to\overline M$ be an immersion of a manifold $M$ into a Riemannian manifold $(\overline M, \overline g)$ and let $H\in C^\infty(M)$ be a given function. I would like to find a functional $I_H(...
Pitollo's user avatar
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2 answers
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Showing that $\int\int |x-y|^{-t}d\mu(x)d\mu(y)<\infty$ provided that $\mu(\mathbb{R}^n)<\infty,\mu(B(x,r))<cr^s, s>t>0$ with $\mu$ a Radon measure

Let $t > 0$ and $\mu$ be a generic Radon measure on $\mathbb{R}^n$ and write $$I := \int_{\mathbb{R}^n}\int_{\mathbb{R}^n}|x-y|^{-t}d\mu(x)d\mu(y)$$ Mattilas's Geometry of Sets and Measures in ...
Cartesian Bear's user avatar
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1 answer
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Do countable collections of points in $B^m$ have Hausdorff 1-dimensional measure zero?

Let $\mu$ denote that Hausdorff $1$-dimensional measure. Suppose $S\subset\mathbb{B}^m$, where $B^m$ is the unit ball in $\mathbb{R}^m$, $m\geq 3$. I'm not so familiar with the Hausdorff measure, but ...
Benjamin's user avatar
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Co-area formula and compositions

I'm working on a problem and came across an integral which seems to involve the co-area formula and I wanted to see if what I'm doing is correct. Let's say we have an integrable function $F:\mathbb{R} ...
brighton's user avatar
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2 votes
0 answers
37 views

Smooth approximation of BV functions, Proof clarification

Im reading Evans&Gariepy book measure theory and fine properties of functions second edition. I have a question about the proof of theorem 5.3 that I don't understand. The theorem states that for ...
Franlezana's user avatar
2 votes
1 answer
130 views

Intuition of minimal surfaces in the class of sets of locally finite perimeter.

I'm reading the book Minimal Surfaces and Functions of Bounded Variation by Enrico Giusti. I'm wondering if anyone could help me understand the intuition of the theorem below. We define $$\...
Franlezana's user avatar
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Integral inequalities with total variation measure

I know and I can prove that, given $f:\Omega \rightarrow \mathbb{R}$ in $L^1_{|\mu|}(\Omega)$, then $$\left|\int_{\Omega}f\,d\mu\right|\leq \int_{\Omega}|f|d|\mu|$$ where $|\mu|$ is the total ...
nimaba99's user avatar
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112 views

When does a random geometric graph become connected?

Fix $n\in \mathbb N$ and let $X_1,\dots,X_n$ be i.i.d uniform random points in $[0,1]^2$. For $r\in \mathbb R$ consider the (random) geometric graph $\mathcal G _r(X)$ with vertices $X=\{X_i\}$ and ...
Alex's user avatar
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1 answer
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A question on the Hausdorff dimension of a subset of $\mathbb{R}.$ [closed]

Let $p\in [0,1].$ I am interested in showing that there exist sets $A,B\subset \mathbb{R}$ of Hausdorff dimension $p$ such that the $p$-dimensional Hausdorff measures $H_p(A)=\infty$ and $H_p(B)=0$. I'...
neophyte's user avatar
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1 vote
1 answer
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Proving a criteria for weak compactness of Radon measures

I'm trying to prove the following criteria for weak compactness of Radon measures: Let $\{\mu_k\}_{k \in \mathbb{N}}$ be a sequence of radon measures on $\mathbb{R}^n$ satisfying $$\sup_{k \in \...
Matheus Andrade's user avatar
1 vote
0 answers
41 views

An isoperimetric-type inequality

I am reading some notes on de Giorgi's methods in the regularity of elliptic equations, and have come across a step which I can't make sense of. The claim is as follows (see Lemma 10 in the linked ...
strtlmp's user avatar
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2 votes
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zero set of a bounded variation function on $\mathbb{R}^3$

Suppose $f(t,s,u)$ is a continuous function on $[0,1]^3$. Suppose $f$ is monotone with respect to $t$, and monotone with respect to $u$. Suppose for each $t,u$ fixed, $f$ is of bounded variation $\leq ...
asdf's user avatar
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1 answer
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Hausdorff dimension of smooth Riemannian manifold is same as its dimension

This came as a discussion of this question Local isometry and Hausdorff dimension . I have looked into this answer Hausdorff Dimension of a manifold of dimension n? but I understood it vaguely. I am ...
User 11111's user avatar
1 vote
1 answer
104 views

Local isometry and Hausdorff dimension

I am currently reading Falconer's book on Hausdorff dimension. My question is whether Hausdorff dimension is invariant under local isometry between smooth Riemannian manifolds? I think it should be ...
User 11111's user avatar
5 votes
1 answer
338 views

Confusion on integration by parts on a Riemannian manifold

For two vector fields $X$ and $Y$ on a Riemannian manifold $M$ with metric $g$, we define $$\langle X, Y \rangle_{L^2} = \int_M g_{ij}X^iY^j dV.$$ I have been unable to find a similar expression for ...
CBBAM's user avatar
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1 vote
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Manifold measure as the limit of "weighted" Lebesgue measures.

Let $\mathcal{M} \subseteq (0,1)^n \subseteq \mathbb{R}^n$ be a d dimensional differentiable manifold and $f: [0,1]^n \rightarrow \mathbb{R}^{n-d}$ be a smooth function such that $\mathcal{M} = f^{-1}(...
Lukas's user avatar
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1 vote
1 answer
31 views

Existence and uniqueneess of functional maximizing value on specific element of $L^p$

Given a function $f\in L^p$ with $1<p<\infty$ and $\|f\|_p=1$ is it true that there is a unique function $g \in L^q$ with $\|g\|_q=1$ such that $\int fg =1$? This is over an arbitrary measure ...
i like math's user avatar
  • 1,063
1 vote
0 answers
61 views

Hausdorff dimension calculation for groups

I was reading about homogeneous space i.e let $G$ be a Lie group and $\Gamma$ be a discrete subgroup of $G$ with finite covolume. Then Hausdroff dimension of $G / \Gamma$ is the Hausdorff dimension of ...
User11111's user avatar
  • 170
5 votes
2 answers
287 views

Bounding $L^1$ norm of the difference between a function $f:\mathbb R^n\to\mathbb R$ of bounded variation and a piecewise constant approximation

As a follow up to this question, which deals with univariate functions, I assume that we are given a function $f:\mathbb R^n\to\mathbb R$ which is of bounded variation on bounded sets, meaning, ...
Václav Mordvinov's user avatar
4 votes
0 answers
105 views

Do there exist continuous maps from the Sphere to the Plane which preserve the Hausdorff Measure of all sets with some Dimension between 1 and 2?

It's well known that there do exist area preserving maps between the sphere and the plane. It's well known there do NOT exist distance preserving maps between the sphere and the plane. So naturally ...
Sidharth Ghoshal's user avatar
5 votes
1 answer
193 views

The Lebesgue measures of open sets of points of epsilon distance from a compact set converge to the Lebesgue measure of that compact set

I’ve found a few questions on this website relating to this question, but none that answer this specific question directly. Let $E$ be a compact subset of $\mathbb{R}$ with Lebesgue measure $\lambda(E)...
no lemon no melon's user avatar

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