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 ...

learn more… | top users | synonyms

10
votes
0answers
348 views

Open problems in Federer's Geometric Measure Theory

I wanted to know if the problems mentionned in this book are solved. More specifically, at some places, the author says that he doesn't know the answer, for example :"I do not know whether this ...
8
votes
0answers
158 views

Higher Order Coarea Formula

I was wondering, if there is a generalization of the coarea formula to higher order derivatives, which would allow one, for example, to relate the norm of the Hessian of a real-valued function $u$ to ...
7
votes
0answers
151 views

Manifolds with volume forms on every submanifold

If we equip a manifold with an inner product (i.e. we have a Riemannian Manifold) then we get a canonical volume form on that manifold (please mentally insert the prefix "pseudo" into my question ...
6
votes
0answers
128 views

How difficult is it to impose a differential structure on a fractal?

Assuming we have a 4-dimensional smooth manifold $M$ embedded in $\mathbb{R}^{m}$ that is difficult to understand its differential structure. For convenience's sake we can assume it is compact, simply ...
6
votes
0answers
306 views

Ham sandwich theorem for integrable functions?

The classical ham sandwich theorem says that given $n$ measurable sets in $\mathbb{R}^n$, it is possible to divide all of them in half (with respect to their measure, i.e. volume) with a single $(n − ...
5
votes
0answers
168 views

Lebesgue density theorem for compact metric spaces.

Let $X$ be a compact metric space (with balls $B_{\varepsilon }(x)$), $\mu $ a Borel probability measure, and $A$ a Borel set with positive probability. Do we have that $\lim_{\varepsilon \...
5
votes
0answers
261 views

Lebesgue measure as a fixpoint: change of variables formulas

This question is inspired by several others on a similar topic: see e.g. this one and a sequence of linked questions. Let us so far focus on $\Bbb R^n$ endowed with standard Borel structure. For any ...
5
votes
0answers
154 views

Measurability of a certain set in Falcolner's Geometry of Fractal Sets

On page 24 of Falcolner's The Geometry of Fractal Sets, Falcolner defines the set $F = \{ x \in E : \mathcal{H}^s(E \cap U) < \alpha$ diam$(U)^s$, for all convex sets $U$ containing $x$ such that $...
5
votes
0answers
259 views

Fourier dimension of a measure restricted to an open set

Suppose that the measure $\mu$ on $\mathbb{R}^n$ has Fourier dimension $\beta$, which is to say that \begin{equation*} \beta= \sup\left\{\gamma \leq n : |\hat{\mu}(x)| \leq C(1+|x|)^{-\gamma/2}\right\...
4
votes
0answers
66 views

Singular points of one-to-one mapping on rectifiable set

Let $E$ be an $s$-rectifiable set in $\mathbb{R}^n$ of positive $s$-dimensional Hausdorff measure $H^s(E)>0$. The original question I have is: Can there exist a one-to-one Lipschitz function $f\...
4
votes
0answers
135 views

Lebesgue Decomposition Theorem only true for Borel sets?

In Evan's book "Geometric Measure Theory and Fine Properties of Functions", we have the following two theorems: Differentiation Theorem for Radon measures. Let $\nu, \mu: \mathcal P(\Bbb R^n) \to [...
4
votes
0answers
43 views

Wasserstein space of order 2.

I have a question about Wasserstein space. I am just wondering if the following statement is true Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space with $\bar{\Omega}$ being a Polish space ...
4
votes
0answers
86 views

Co-Area formula in Riemannian geometry

I wonder if the following holds true: Let $z:[0,1]\times B^{n-1}_r(0)\to(M^n,g), (t,p)\mapsto z(t,p)$ be a diffeomorphism a.e. onto its image with respect to the Lebesgue measure on $A:=[0,1]\times B^{...
4
votes
0answers
70 views

BV Function times characterstic function still BV?

I am wondering: given a function $u \in BV(\Theta)$ where $\Theta$ is an open subset in $\mathbb{R}^n$ and a Borel subset $B \in \mathcal{B(\Theta)},$ is the function $w \colon= u \chi_B$ still in $BV(...
4
votes
0answers
168 views

Alternative rigorous definition of a surface integral

Consider some open subset $U$ of $\mathbb{R}^n$ where $U$ has a (piecewise) $C^1$-boundary. Let $f$ be some smooth (enough) real function. Is there some way to give a measure-theoretic definition of ...
4
votes
0answers
125 views

Limits to the growth of the volume of a union of spheres

Assume that $x_i$, $i=1,\ldots,m$ are points in $\mathbb{R}^n$, with the maximal distance between any two of them being at most $1$. Define $$ a(r)=\mu\Bigl(\bigcup_{i=1}^m B(x_i,r)\Bigr),$$ where $\...
4
votes
0answers
217 views

Three properties of the Lebesgue measure on $\mathbb{R}^n$

I'm writing notes for my upcoming class in Game Theory and I realized some time ago that I only need three properties of the Lebesgue measure $\lambda$ on $\mathbb{R^n}$. It is a non-negative ...
4
votes
0answers
174 views

Integral Geometry Reference Request

I am looking for a good introductory reference (book, lecture notes, survey article) on integral geometry. I am especially interested in the Crofton formula in $\mathbb{R}^n$ and its extensions to ...
3
votes
0answers
50 views

Constructing a null set and a Lipschitz function nowhere differentiable on it

I'm trying to solve the following exercise. Now, Rademacher's theorem says that locally Lipschitz functions are $\mathcal L^N$-a.e. differentiable, so $E$ must be a null set, and this is clearly ...
3
votes
0answers
388 views

Is a Jordan curve encircling a finite-perimeter set rectifiable?

Let $\gamma:[0,1]\rightarrow \mathbb R^2$ be a (continuous) simple closed curve (Jordan curve). The curve is not assumed to be rectifiable, i.e. we don't assume a priori that the length of the curve $$...
3
votes
0answers
127 views

Can you determine the length of a curve by the lengths of its projections onto planes?

If $\,\Gamma \subset \mathbb R^n$ is $1$–rectifiable, then its Hausdorff measure is equal to its integral geometric measure. That is, $$\displaystyle\mathcal H^1\left(\Gamma\right) = \int_{G\left({1,\...
3
votes
0answers
62 views

The projection of density 1 point on a rectifiable set.

This post has also been posted here. Please see the comment on the linked page, useful information! Let $\Gamma\subset \mathbb R^N$ be $\mathcal H^{N-1}$-rectifiable. Then we know that $\mathcal H^{N-...
3
votes
0answers
43 views

When a current is actually a holomorphic form?

If a current $f$ of bidegree $(p,0)$ (acting on forms of bidegree $(n-p,n)$) satisfies $\bar{d}f=0$, is it true that $f$ is a holomorphic differential form? In general, do we have any standard ...
3
votes
0answers
55 views

Sobolev Multiplication theorem for Fibre bundles

Let $X$ be a compact, oriented, four dimensional Riemannian manifold and $Q\longrightarrow X$ be a principal $G$-bundle over $X$ for a smooth, compact Lie group $G$. Let $M$ be a manifold admitting a ...
3
votes
0answers
50 views

A covering argument for metric Jacobian

Given a Lipschitz map between Carnot Groups $ f : G_1 \to G_2$, with homogeneous dilations $ \delta^1_s, \delta^2_s$, we have the almost everywhere Pansu derivative $ D_H f(x)(y) = \lim_{s\to 0} \...
3
votes
0answers
78 views

Density and $\lambda$ - measurability of a Radon measure

Question: Let $\lambda$ be a Radon measure on $\mathbb{R}^n$ and $A \subset \mathbb{R}^n$. Show that $$ \lim_{r \rightarrow 0} \frac{\lambda(A \cap B(x,r))}{\lambda(B(x,r))}=0 \ \ \ \text{for $\...
3
votes
0answers
84 views

Measurability of points regular

I'm reviewing the proof of the theorem of oseledet the book Mañe: Let $M$ a compact metric space and $f:M \rightarrow M$ a homeomorphism, $\pi: F \rightarrow M$ a finite-dimensional continuos vector ...
3
votes
0answers
201 views

Co-area formula involving non-integer Hausdorff measure

Is there any co-area formula involving non-integer Hausdorff dimension? Moreover is it sensible to write the following: Let $S$ be a subset in $ \mathbb{R}^n$ with Hausdorff dimension s $(0<s<n)$...
2
votes
0answers
91 views

Area and coarea formula and its application, change of variables

I have a question about an application of area and coarea formula and change of variables. Let $D$ be a bounded and connected open subset of $\mathbb{R}^{d}$ with $C^{1}$-boundary. Define $D_{\...
2
votes
0answers
34 views

Polar coordinates, bounded domain with $C^{1}$ boundary

I have a question about a integral on a surface. It is well known that for any Integrable function $f$ defined on $\mathbb{R}^{n}$, it holds that \begin{equation} (1) \quad \frac{d}{dr} \int_{B(0,r)}...
2
votes
0answers
67 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 $BV(\Omega)$. ...
2
votes
0answers
22 views

Estimating Self-Convolution of Surface Measure on Sphere

Let $n\geq 2$, and let $\sigma$ denote the standard surface measure on the $n$-sphere $S^{n-1}$, normalized to have total measure $1$. According to the exercise at the end of Terence Tao's blog post ...
2
votes
0answers
58 views

Approximation of Sets of Finite Perimeter

Fix an open set $\Omega \subset \mathbb{R}^n$. If $E$ is a measurable subset of $\Omega$, we may define the perimeter of $E$ in $\Omega$, denoted by $P(E;\Omega)$, to be $$P(E;\Omega) = \sup_{\...
2
votes
0answers
54 views

Image of a Jordan compact set under a degenerate map

Briefly: Is the image of a Jordan compact set $K$ under a degenerate smooth map $\varphi$ equal to an image of a compact subset $T\subseteq K$ of zero measure, $\mu(T)=0$: $$ \varphi(K)=\varphi(...
2
votes
0answers
41 views

Blow-up of derivative of BV function at the jump set

"Motivation" Let $u\in BV(\mathbb{R}^n)$ be a function of bounded variation, and let $x\in J_u$ be a point in its jump set. For $\mathcal{H}^{n-1}$-a.e. such $x$, we can define the unit normal $\nu$ ...
2
votes
0answers
305 views

The Co-area formula for $BV$ function V.S. the co-area formula for $C^\infty$ functions

I am working on the proof of the co-area formula for $BV$ functions. Suppose $u\in BV(\Omega)$ then the co-area formula states that $$ \|Du\|(\Omega)=\int_{-\infty}^\infty \|\partial E_t(u)\|(\Omega)\...
2
votes
0answers
103 views

Crofton formula in higher dimension

In the plane, the Crofton formula states that for a rectifiable plane curve $\gamma$, we have $\int |line \cap \gamma| d\Omega_1=2\times length(\gamma)$ where $d\Omega_1$ is the translation/rotation-...
2
votes
0answers
58 views

Difference of two subharmonic functions and signed measures

One of the reasons subharmonic functions are interesting is that if you take their laplacian, you get a measure (and conversely any finite Radon measure with compact support can be obtained in this ...
2
votes
0answers
88 views

Problem with definition of pushforward density

Let $X$, $Y$ be smooth manifolds and let $\pi \colon X \to Y$ be a submersion. Then for every $y \in Y$ the set $W_y = \pi^{-1}(y)$ is a submanifold in $X$. Let $\mu \in \Gamma(|\Lambda| X)$ be a ...
2
votes
0answers
74 views

$L^1$ approximate continuity points of epigraph of a continuous function

Consider a continuous function $f\colon \mathbb R\to \mathbb R$. Let $F(x,y):=\mathrm{sign\,}(y-f(x))$. Is it true that for a.e. $x\in \mathbb R$ the point $(x,f(x))$ is not a point of $L^1$ ...
2
votes
0answers
74 views

Measures whose projections are absolutely continuous

Let $\mu$ be a finite Borel measure on the plane. Does there exist a characterization of the property that almost all (wrt rotations) projections of $\mu$ to lines on the plane are absolutely ...
1
vote
0answers
27 views

Can we always extract a proper Hausdorff measurable subset from a Hausdorff measurable set?

This question has been post in here. Let $\Gamma\subset \Omega\subset \mathbb R^N$ be such that $\mathcal H^{N-1}(\Gamma)<+\infty$ (this also implise that $\Gamma$ is Hausdorff measurable). ...
1
vote
0answers
20 views

Help understanding proof of Frostman's Lemma - issue technical or termonological?

I was reading Hochman's proof of Frostman's lemma in his online lecture notes here and got hung up. I'm not sure if I'm missing a part of the proof or I'm misunderstanding the theorem itself. The ...
1
vote
0answers
17 views

Bounding below Hausdorff measure of conected set

I'm trying to prove that for every connected set $E\subset\mathbb{R}$, $H^1(E)$, the Hausdorff measure is bounded below by $\text{diam}(E)$. In the answer there, it was suggested to use a Lipschitz ...
1
vote
0answers
19 views

Identification of the normal cycle to a closed curve

I'm studying the normal cycles in Morvan's book "Generalized curvatures"; He says that the normal cycle of a domain D bounded by a smooth curve is the current ( i.e the linear continuous functional ...
1
vote
0answers
35 views

approximate $C^1$ function by Holder function

I am trying to prove the following Lemma: Suppose $u$: $\mathcal R^N\to \mathbb R$ is $C^1$. Then for each $\epsilon>0$, there exists a $C^{1,\alpha}$ function $\tilde f$ such that $$ \...
1
vote
0answers
25 views

Topologies on Spaces of k-rectifiable sets, and $C^k$ convergence

I've been thinking about topologies on the spaces of k-rectifiable sets (Hausdorff metric topology, varifold topology, etc) in $\mathbb{R}^n$, and I'm wondering if there are hypotheses under which ...
1
vote
0answers
39 views

Hausdorff Measure under linear maps

We have a linear map $L : \mathbb R^n \rightarrow \mathbb R^m $ ($m\geq n$) I have two questions: How does one prove that $\mathcal H^n (L(B(x,r)))=\mathcal L^n (O^* \circ L(B(x,r)))$? (We have ...
1
vote
0answers
8 views

$\mathbf{M}$ is $\mathbf{F}$ lower semicontinuous on $\mathscr{D}_m$

I am working on a problem in Frank Morgan's Geometric Measure Theory book. What I have done so far: $T_i \to T$ under the real flat norm: $\forall \epsilon>0, \exists N$ such that $\min \{ \...
1
vote
0answers
49 views

Structure theorem for locally bounded variation functions

I have some doubts reading Measure Theory and Fine Properties of Functions by Evans and Gariepy. In particular, they define the space of locally bounded variation functions $BV_{loc}(U)$ in $U\subset \...