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.
741
questions
10
votes
1
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953
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A strange Lipschitz function
Let $n \geq 3$. Does there exist a Lipschitz function $f: \mathbb R^n \to \mathbb R$ such that the following conditions hold?
The origin is a weak Lebesgue point of $\nabla f$, in the sense that the ...
- 6,298
2
votes
2
answers
145
views
Domains of type (A) are Lipschitz?
In this article and in the book of Ladyzhenskaya et al (1968) - Linear and Quasilinear Elliptic Equations we have the following definition of what is a domain of type (A):
There is no example of a ...
- 1,759
2
votes
0
answers
69
views
Connectedness of Space of Caccioppoli Sets?
Let $(M^n, g)$ a closed, connected riemannian manifold. Is the space of all caccioppoli sets on $M$ connected with respect to the flat norm? How about with respect to the F norm (Flat metric on the ...
- 339
1
vote
0
answers
67
views
Hausdorff distance and Hausdorff measure of symmetric difference
Let $X_n$ be a sequence of $k$-dimensional piecewise smooth submanifolds of $\mathbb{R}^m$, converging in Hausdorff distance to a $k$-dimensional piecewise smooth submanifold $Y \subset \mathbb{R}^m$, ...
- 11
1
vote
1
answer
77
views
When is a $1$-varifold $V$ the associated varifold of the reduced boundary of some Caccioppoli set?
Let $v_1$, $v_2$, $\cdots$, $v_l\in\mathbb{R}^n$ be unit vectors, $\mathbb{R}_v^+:=\{\lambda v:\lambda>0\}\subset\mathbb{R}^n$ be the ray in $v$'s direction; $n_1$, $n_2$, $\cdots$, $n_l>0$ be ...
- 111
2
votes
0
answers
28
views
Steiner symmetrization of smooth function on non-simply connected regions
Given a smooth function $u$ defined on $\mathbb{R}^2$, restrict $u$ to a subset $\Omega \subset \mathbb{R}^2$ (possibly not simply connected) foliated by level sets of a smooth function $\psi: \Omega \...
- 434
3
votes
1
answer
169
views
Probability measure on partition theorem
Probability measure in $\mathbb{R}^1$: Continuous function $f$ that is positive everywhere $\ \int_{-\infty}^{\infty} f dx =1$ (total area under the curve equals 1). For visualization see here.
...
- 9
7
votes
1
answer
136
views
Higher (BV) regularity of solutions to Poisson equation with Radon measure right-hand side?
I am trying to understand higher regularity for solutions to Poisson's equation when the right-hand side is a Radon measure. In particular:
$$\begin{cases}
\Delta u = \mu \text{ in } \Omega\\
u = 0\...
- 221
1
vote
2
answers
88
views
Ratio of Gaussian measure over Euclidean balls
Let $\nu\in\mathcal P(\mathbf R^d)$ be the standard Gaussian distribution $\mathcal N(0,I_d)$.
Denote by $\mathscr B$ the class of Euclidean balls $B_r(x)$ (centered in $x\in\mathbf R^d$ with radius $...
- 11
1
vote
1
answer
194
views
A question on Borel measurability
Let $(X, \mathcal{B}_{X}, \mu)$ be a measure space. Here, $\mu$ is an infinite Borel measure and $\mu$ is not $\sigma$-finite. Let $\pi$ be surjective Borel measurable map form $(X, \mathcal{B}_{X}, \...
- 11
5
votes
1
answer
208
views
Intersection between Lipschitz domains
Let $\Omega\subset\mathbb{R}^N$ be an open, bounded and connected Lipschitz domain. Is it true that we can find some $R>0$ such that any $N$-dimensional open ball $B(x,r)$ with $r\leq R$ that ...
- 1,759
8
votes
0
answers
395
views
For $f$ Lipschitz with $|\nabla f| = 1$ a.e., what is the supremal Hausdorff dimension of the set on which $\varepsilon< |\nabla f| < 1-\varepsilon$?
Let $f$ be a Lipschitz function with $|\nabla f| = 1$ almost everywhere.
Let $\varepsilon \geq 0$. What is the supremal Hausdorff dimension of the set on which $f$ is differentiable with $\varepsilon &...
- 6,298
3
votes
1
answer
226
views
Can any function in $C^\alpha$ be approximated in $C^{\alpha^-}$ by singular functions?
For every positive $\alpha < 1$, we consider the space $C^{\alpha}$ of Holder continuous functions of order $\alpha$ on $[0, 1]$, equipped with the norm
$$\|f\|_{C^\alpha} := \sup|f| + \sup_{x, y \...
- 6,298
3
votes
1
answer
244
views
Hausdorff dimension of the zero set of the gradient of an eikonal function
Let $f: \mathbb R^n \to \mathbb R$ be a Lipschitz function with $|\nabla f| = 1$ almost everywhere with respect to Lebesgue measure.
What is the supremal Hausdorff dimension of the set on which $f$ is ...
- 6,298
5
votes
2
answers
236
views
Hausdorff dimension of the zero set of $\nabla f$
Let $f: \mathbb R^n \to \mathbb R$ be a Lipschitz function with $\nabla f$ nonzero almost everywhere with respect to Lebesgue measure.
What is the supremal Hausdorff dimension of the set on which $f$ ...
- 6,298
9
votes
1
answer
400
views
Dispersion points of Lipschitz functions
For a function $f: \mathbb R^n \to \mathbb R^m$ with $m < n$, we say that $x \in \mathbb R^n$ is a dispersion point of $f$ if
$$\liminf_{y \to x} \frac{|f(y) - f(x)|}{|y - x|} > 0.$$
Question: ...
- 6,298
3
votes
1
answer
97
views
How to check that the surface measure is the weak limit of $\delta^{-1}\mathcal{L}^n|_{B(0,1+\delta)\setminus B(0,1)}$?
We denote the unit sphere $\{x\in\mathbb{R}^n:|x|=1\}$ by $S^{n-1}.$ If $x\in\mathbb{R}^n\setminus\{0\}$, the polar coordinates of $x$ are
\begin{align*}
r=|x|\in(0,\infty),\quad \gamma=\dfrac x{|...
- 207
6
votes
1
answer
711
views
Is the $L^\infty$ norm of the derivative the same under the Hausdorff and Lebesgue measure?
Note: Here $\mathcal H^k$ denotes the $k$-dimensional Hausdorff measure, and $\|f\|_{L^\infty (\mathcal H^k)}$ denotes the $L^\infty$ norm of a function $f$ with respect to $\mathcal H^k$.
Let $\Omega$...
- 6,298
5
votes
1
answer
229
views
Are singular functions dense in the space of Hölder continuous functions?
We say a non-constant function $f$ on $[0, 1]$ is singular if it is continuous, and in addition differentiable almost everywhere with $f' = 0$ a.e.
For every positive $\alpha < 1$, is the set of ...
- 6,298
4
votes
1
answer
322
views
Reference request: Intuitive introduction to currents and varifolds
As I have recently been interested in geometric measure theory related problems, I am learning some of the basics of the field. I am looking for a textbook that introduces currents and varifolds in an ...
7
votes
2
answers
440
views
Uncountable collections of distinct subsets of an interval (existence)
Throughout, $\mu$ is just the Lebesgue measure.
Question: does there exist an uncountable family of distinct subsets of $[-1, 1]$, denoted by $(U_j)_{j \in [-1, 1]}$, with $\mu(U_j) > 0$ for each $...
- 101
5
votes
1
answer
251
views
Is there a singular function that is Hölder continuous of every order less than $1$?
We say a non-constant function $f$ on $[0, 1]$ is singular if it is continuous, and in addition differentiable almost everywhere with $f' = 0$ a.e.
Does there exist a singular function that is Hölder ...
- 6,298
16
votes
4
answers
2k
views
Is the $W^{1, \infty}$ limit of differentiable functions also differentiable?
Let $f_n$ be a sequence of differentiable functions on $[0, 1]$ with
$f_n \to f$ uniformly for some (necessarily) continuous $f$.
$f'_n - g \to 0$ in $L^{\infty}$ for some measurable $g$.
Is it true ...
- 6,298
5
votes
2
answers
284
views
Is the $W^{1, \infty}$ limit of differentiable a.e. functions also differentiable a.e.?
Let $f_n$ be a sequence of continuous, differentiable a.e. functions on $[0, 1]$ with
$f_n \to f$ uniformly for some continuous $f$.
$f'_n - g \to 0$ in $L^\infty$ for some measurable $g$,
where we ...
- 6,298
6
votes
1
answer
276
views
Is the derivative of a $C^1$ function nonzero almost everywhere on almost every level set?
Note: Here $\mathcal H^k$ denotes the $k$-dimensional Hausdorff measure.
Let $f \in C^1 (\mathbb \Omega)$ for some open, connected, bounded subset $\Omega$ of $\mathbb R^n$. We consider for each $t \...
- 6,298
3
votes
0
answers
60
views
How powerful are sequences of Steiner symmetrizations?
I was studying geometric analysis and have encountered something called Steiner symmetrization method. Intuitively I understand how it's made to be applied and used, but Wikipedia pages do not give ...
- 131
1
vote
1
answer
162
views
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 ...
- 119
0
votes
0
answers
64
views
Wasserstein space isomorphic to original space?
Is there a complete measurable metric space $(X,d)$ for which its $p$-Wasserstein space $W(X)$ is isometrically isomorphic to $(X,d)$ for some $p \in [1,\infty]$?
Note that there is a canonical non-...
1
vote
0
answers
55
views
Boundary behavior for submanifolds with bounded second fundamental form
I am interested in a boundary version of this question About hypersurfaces in R^n+1 with bounded 2nd fundamental form.
The question is as follows. Let $\Sigma^k\subset \Bbb R^n$ be a submanifold with ...
- 151
1
vote
1
answer
168
views
For proper group action on closed Riemannian manifold, must the union of orbits with non-unique closest points to a given point be of 0 volume measure
Let $(M,g)$ be a closed (compact without boundary) Riemannian manifold of finite dimension, with the volume measure $\mu:= \mu(E):=\int_{E}d\operatorname{vol}_g \forall E \in \mathcal{B}(M),$ the ...
- 1,427
3
votes
1
answer
193
views
Product of low dimensional Hausdorff measures
Let $\mathcal{H}^n$ and $\mathcal{H}^m$ be Hausdorff measures on $\mathbb{R}^n$ and $\mathbb{R}^m$. We know that the product measure $\mathcal{H}^n\otimes \mathcal{H}^m$ is the Hausdorff measure $\...
- 73
0
votes
1
answer
66
views
Hahn-Mazurkiewicz with finite one-dimensional Hausdorff measure
Suppose that there is a continuous surjection from $[0,1]$ to a metric space $(X,d)$. If $(X,d)$ has finite one-dimensional Hausdorff measure, must there exist a Lipschitz surjection from $[0,1]$ to $(...
- 133
6
votes
1
answer
338
views
Densities, pseudoforms, absolute differential forms and measures, differential forms, etc
Apologies if this question is too basic, but I figured I first heard of most of these concepts on MO, so perhaps I can ask here.
Gelfand’s definition, copied from AlvarezPaiva [My edit, could be ...
- 771
2
votes
1
answer
119
views
On the existence of a complicated fractal-like set of finite perimeter
Let $f\in BV(\Bbb R^n)$ be an integer-valued function that maps into $\{0, 1\}$ and is identically $0$ outside some bounded set in $\Bbb R^n$. In particular, $f$ determines a bounded Caccioppoli set $...
- 1,245
5
votes
1
answer
217
views
Solution to the Eikonal equation with almost everywhere continuous derivative
Let $\Omega$ be an open, bounded, connected subset of $\mathbb R^n$ with smooth boundary. Does there always exist an almost everywhere solution $u \in W^{1, \infty}$ to the following system of PDE?
$$|...
- 6,298
2
votes
0
answers
185
views
Statistical invariants of Riemannian manifolds
$\DeclareMathOperator\diam{diam}\DeclareMathOperator\rad{rad}\DeclareMathOperator\iso{iso}\DeclareMathOperator\com{com}\DeclareMathOperator\con{con}$A cheap way of defining invariants of Riemannian ...
- 139
0
votes
0
answers
32
views
On the Lipschitz parametrizability of polynomials of fixed Mahler measure
Background
For a polynomial $f(x) = a(x-\alpha_1) \cdots (x - \alpha_n) \in \mathbb{C}[x]$, its Mahler measure is defined to be
$$M(f) = |a| \prod_{i=1}^n \max\{1, |\alpha_i|\}$$
In Lemma 1, Masser-...
- 267
0
votes
0
answers
88
views
Why does $\omega$ belong to $D^{\varepsilon^3}\theta^\ast$ for $\lesssim \rho^{-n} D^{n-2 + n\varepsilon^3}$ different $\theta$?
In this paper, there is the following claim (Pg. 1850):
If $1 - \eta(w) \ne 0$, then $|\omega| \ge \rho$. In that case, $\omega$ belongs to $D^{\varepsilon^3}\theta^\ast$ for $\lesssim \rho^{-n} D^{n-...
- 165
2
votes
0
answers
88
views
Equivalence class of parametrized surfaces which induce the same current
Suppose $M$ is a smooth manifold of dimension $n \geq 2$. A $k$-current is a linear functional on compactly supported smooth forms on $M$, denoted $T: \Omega^k_c(M) \to \mathbb{R}$.
Let $X: [0,1]^2 \...
- 21
1
vote
0
answers
56
views
Proof that, for $u \in H^1$, $\{ u > \alpha \}$ is a quasi open set
I am reading the monograph by A. Henrot, Extremum problems for eigenvalues of elliptic operators. In chapter 2, the notion of a quasi-open set is defined (see the relevant definitions at the end of ...
- 479
2
votes
0
answers
85
views
Measurability of the union of cut loci along a curve
Let $(M,g)$ be a Riemannian symmetric space and $\alpha(s)$ be a geodesic. Define
$$
U(t)=\cup_{s\in[0,t]}{\rm Cut}(\alpha(s))
$$
as the union of the cut loci ${\rm Cut}(\alpha(s))$ along the curve $\...
- 125
1
vote
1
answer
156
views
Differentiability of an integral of geodesic distance
Let $(M,g)$ be an $m$-dimensional Riemannian symmetric space and $d(\cdot,\cdot)$ be the geodesic distance function. Fix any $\alpha\in M$ and $v\in T_\alpha M$ with $\|v\|=1$.
Q1: Define
$$
g(t)=\...
- 125
0
votes
0
answers
94
views
Bounding the area of the image of a set by product of maximum of lengths
Let $F:[0,1]\times[0,1]\to \mathbb {R}^2$ be a smooth function. Given $x\in [0,1]$, let $\ell_x:=\{x\}\times [0,1]$, and given $y\in [0,1]$, let $\ell_y:=[0,1]\times \{y\}$.
My question feels ...
- 503
4
votes
1
answer
191
views
What are the possible blow up limits of an $L^1$ function?
Let $f: [0, 1] \to \mathbb R$ be an $L^1$ function. Define for each $r > 0$, the blow up $f_r:[0, 1] \to \mathbb R$ by
$$f_r (x) := \frac{f(rx)}{r}.$$
Suppose $f_r$ converges in $L^1$ to some ...
- 6,298
2
votes
0
answers
150
views
$\mathscr{H}^{n-2}(\Sigma)< \infty$ implies $\mathscr{H}^{n-1}(\pi(\Sigma))=0$
Let $\Sigma\subset \mathbb{R}^{n+1}$ be a set with $(n-2)$-dimensional Hausdorff measure finite, i.e. $\mathscr{H}^{n-2}(\Sigma)<\infty$. Let $\pi:\mathbb{R}^{n+1}\to \mathbb{R}^n$ be the ...
- 1,139
2
votes
1
answer
205
views
Macroscopic sets - a notion of largeness for Lebesgue null sets
Let $E$ be a measurable subset of $\mathbb R$. We say $E$ is $\alpha$-macroscopic, for $0 \leq \alpha \leq 1$, if there exists an $\alpha$-Holder continuous function $f: \mathbb R \to \mathbb R$ such ...
- 6,298
1
vote
0
answers
90
views
Periodic orbits in planar smooth billiard table with large periods
Given a plane billiard table with a smooth boundary which is a Jordan curve, I wonder if there is always a periodic orbit with sufficiently large period.
Formulation of my question: We are considering ...
- 429
0
votes
0
answers
92
views
When can an affine functional on the dual be represented as an element of a Banach space?
In Measures Which Agree on Balls by Hoffmann-Jørgenson, we are given a functional $\varphi: T(x_0)\to (-\infty, \infty]$, which is a lower semicontinuous, affine, Baire function on a subspace $T(x_0)$ ...
- 297
1
vote
2
answers
114
views
Computation of tangent functional
In Measures Which Agree on Balls by Hoffmann-Jørgenson, the tangent functional is defined as follows.
If $x \in S$, we define the tangent functional $\tau(x,\cdot)$ at $x$ as
\begin{equation}
\...
- 297
1
vote
1
answer
287
views
Gateaux differentiability of the norm in Banach spaces
I'm struggling to understand a particular implication in the proof of Corollary 5 of this paper involving Gateaux differentiability of the norm. The claim is that Gateaux differentiability of the norm ...
- 297