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.
763 questions
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$L^p$ function whose graph is not a varifold
I am looking for an example of a function $f:[0,1]\to\mathbb{R}$ which is in $L^p$ for some $p$ and whose graph is not a $1$-dimensional varifold in $\mathbb{R}^2$, that is such that it is not ...
- 1,149
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389
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Coarea formula for measure of epsilon neighbourhood
I am trying to use the coarea formula to get estimates on the measure of an epsilon-neighbourhood of a set. Specificly, given a compact 'nice' set $A\subseteq \mathbb{R^d}$, possibly with more than ...
- 633
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115
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Approximating the probability of a half-space using random Voronoi diagrams
Fix a half-space $H = \{x_1 \geq 0: ~ (x_1,\dots,x_n) \in \mathbb{R}^n\}$. Let $p$ be a distribution with support in $\mathbb{R}^n$. I am interested in the following way of estimating the weight $p(H) ...
- 33
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391
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Does a submanifold of nonzero codimension have measure zero under the product of non atomic measures?
Let $A$ be a non atomic measure on $\mathbb R$. Consider the product measure $\mu := A \times \dots \times A$ on $\mathbb R^n$.
Question: Let $M$ be a $n-1$ dimensional smooth submanifold of $\mathbb ...
- 6,215
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258
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Isoperimetric inequality for domains in the exterior of a precompact open set in Riemannian manifold
Fix $n\geq 2$ and let $$\mathbb{H}^{n}=\mathbb{R}_{+}\times \mathbb{S}^{n-1}$$ be the hyperbolic space, so that any point $x\in \mathbb{H}^{n}$ can be represented in polar coordinates $x=(r, \theta)$, ...
- 459
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201
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Does weak continuity of Jacobians hold for non nondegenerate maps?
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
Let $\M,\N$ be two-dimensional smooth, compact, connected, oriented Riemannian manifolds. (with or without boundaries).
Let $f_n \...
- 6,741
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187
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A Frostman-type result for measures satisfying uniform lower density conditions
Let $\mu$ be a finite, compactly supported, non-zero measure on $\mathbb{R}^d$ for an integer $d$. Let $B(x,r)$ denote the ball of radius $r>0$ centered at $x \in \mathbb{R}^d$. For $\delta \in [0,...
- 11
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146
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Is a locally invertible weak limit of injective maps injective almost everywhere?
This is a cross-post.
Let $\Omega_1,\Omega_2 \subseteq \mathbb R^2$ be open, connected, bounded, with non-empty $C^1$ boundaries.
Let $f_n:\bar\Omega_1 \to \bar\Omega_2$ be Lipschitz injective maps ...
- 6,741
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249
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(Novel?) notion of concentration/dispersion
Consider the measurable space $(\Omega,\mathscr{B})$ endowed with two positive measures: a "volume $\nu$" and a probability measure $\mu$. For example, one might take $\Omega=\mathbb{R}^n$ (with the ...
- 6,541
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Growth assumption and example of finite (arbitrarily small) time blow up for ODE
Consider the following ODE initial value problem
\begin{align*}
&\frac{d}{dt}\Phi(t,x) = \boldsymbol{F}(t,\Phi(t,x)), & t \in [0,T], \ \ x \in \mathbb{R}^N,\\
&\Phi(0,x) = x, & x \in \...
- 839
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169
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Difference quotient for solutions of ODE and Liouville equation
Suppose that $\Phi$ is the solution of
$$\begin{cases}
\frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\
\Phi(x,0) = x \quad x \in \mathbb{R}^N
\end{cases}$$
How does one prove that
$$\...
- 839
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279
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Reference request: Functions of bounded variation in one real variable
Is there a good reference for facts and theorems about BV real valued functions? I’m looking for something with much more than say Stein and Shakarchi 3, or Evans and Gariepy. Thanks!
- 2,069
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218
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From measurable to quantitative estimates of a map in the coarea formula
Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m $ be Lipschitz and $n \geq m$. A version of the coarea formula says:
$$ \int_A g(x) J_m f(x) d \mathcal{L}^n (x) = \int_{\mathbb{R}^m } \int_{ A \cap f^{-...
- 365
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269
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Convergence of an iterated sequence
Let $K=[0,1]^2$ be a square and $p\in (0,1)$ be a fixed number. We define a map $F: K^2\to K^2$ as follows.
For $(x_1,y_1), (x_2,y_2)\in K$, it follows by a straightforward computation that there ...
- 345
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55
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Multiplicity of a subcovering in spaces of given Hausdorff dimension
Let $X$ be a locally compact metric space of integer Hausdorff dimension $n$. Let $K\subset X$ be a compact subset. Let $\{B_i\}_i$ be a finite family of balls covering $K$. One may assume that all ...
- 21.8k
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258
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Aronszajn measure
In a geometric measure theory (GMT) course I'm following this year, the professor told us about the Aronszajn measure, and asked us to go check by ourselves what it reprensents (the course was about ...
- 1,616
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Is the speed of a curve in $ \ell^\infty $ zero a.e. if the derivative of each component is zero a.e.?
Let $ A $ be an $ \mathcal{H}^1$-measurable subset of $ \mathbb{R} $ and $ \gamma \colon A \subseteq \mathbb{R} \to \ell^\infty $ be a Lipschitz mapping with the Lipschitz constant $ L $. Also, assume ...
- 520
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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 ...
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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
$$
g(t)=\...
- 125
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276
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Defining area / n-volume of a finite metric space
Let $(X, d)$ be a finite metric space. I've seen several answers to the question when can $X$ be isometrically embedded into Euclidean space (or, more generally, Riemannian manifold). I'm interested ...
- 820
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125
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How to find the point at minimal average distance of a given measure
Given a compactly supported probability measure $m$ on $\mathbb{R}^n$, we can define its average distance to a point $x$ as $\int_\mathbb{R^n}d(x,y)dm(y)$. In this question I found that for a given ...
- 10.6k
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318
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What is the limit of a helix as the frequency tends to infinity?
Consider the helix parametrized by $r(t) = (\cos(\omega t), \sin(\omega t), t)$, for a given $\omega > 0$, and $t \in \mathbb{R}$. How can we interpret the limit as $\omega \to \infty?$
My initial ...
- 217
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172
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A question about pushforward measures and Peano spaces
Specifically my question is the following: Let $P$ be a Peano space. If $(P,\sigma,\mu)$ and $(P,\sigma,\nu)$ are both nonatomic probability measures, does there exist a continuous function $f:P\to P$ ...
- 33
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140
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Isoperimetric inequality for exterior domains on $\mathbb{H}^{n}$
Fix $n\geq 2$ and let $\mathbb{H}^{n}=\mathbb{R}_{+}\times \mathbb{S}^{n-1}$ be the hyperbolic space be defined as a Riemannian manifold equipped with the Riemannian metric $$g=dr^{2}+\sinh^{2}rd\...
- 459
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213
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Stability of isoperimetric inequality
Let $S$ be subset of $\mathbb{R}^n$ with perimeter 1.
Isoperimetric inequality states that then the volume of $S$ is not greater than $V_n$,
where $V_n$ is the volume of a ball in $\mathbb{R}^n$ with ...
- 2,576
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88
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Convergence of probability measures which (asymptotically) concentrate along a submanifold
Let $V : (-1, 1)^d \to \mathbf{R}_+$ be a smooth function, and for $\beta > 0$, define
\begin{align}
P_\beta ( dx ) &= \exp \left( - \beta V ( x ) \right) / z (\beta) \, dx\\
z (\beta) &= \...
- 801
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273
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How to prove space of non-negative Radon measures is complete?
Let $\mathcal{M}^{+}(\mathbb{R}_{+})$ be space of non-negative Radon measures on $\mathbb{R}_{+}$ with bounded total variation and define the metric $\rho$ on $\mathcal{M}^{+} (\mathbb{R}_{+})$ as $$ \...
- 133
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141
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Approximating measure by partitions
Let $(X,\mathcal{B},\mu)$ be a non-atomic Borel probability space. We may assume that $X\subseteq \mathbb{R}^d$ is the open (or closed) unit ball, if it helps.
Let $\mathcal{C}$ be a countable ...
- 502
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116
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Lipschitz vs. bi-Lipschitz parametrizations for subsets of Euclidean space [closed]
Let $n \in \mathbb{N}$. Is there a standard example of a subset of $\mathbb{R}^{n+1}$ that is contained in the image of a Lipschitz map $\mathbb{R}^n \to \mathbb{R}^{n+1}$ (or, more generally, that is ...
- 565
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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 $\...
- 135
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Let $\mathcal{M}(\Xi)$ set of all probability destributions on $\Xi$. Supremum over $\mathcal{M}(\Xi)$ is equal to sup over Dirac distributions
This doubt is born because I am reading an article in this link in pag 12 in order to use these ideas to prove another problem that raised me. My doubt is following:
Let $(\Xi,\mathcal{E})$ be a ...
- 159
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269
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Existence of stationary tangent cones
My question refers to Leon Simon's Book: Geometric Measure Theory, Chapter 42
So let $V\in G_n(U)$ a general Varifold and $\|\delta V\|$ the Total Variation measure. If the density $\theta^n(\mu_V,x)&...
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Image of a Jordan compact set under a degenerate map
This is crossposted from MSE, I hope this is suitable here, since there is no reaction there. I need this lemma for teaching, and I would appreciate any help.
Briefly:
Is the image of a Jordan ...
- 7,426
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Is it true that $ \mathcal{H}^{n-1} (\operatorname{spt} \mu _E \setminus \partial ^{*}E)=0$?
In Federer's Theorem, $ \mathcal{H}^{n-1} (\partial ^{m}E \setminus \partial ^{*}E)=0 $, where $E$ is a set of finite perimeter in $ \mathbb R^n $, $\partial ^{e}E$ is the essential boundary of E, and ...
- 1,350
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Half-space comparison of perimeter
Claim: suppose that $E$ is a set of finite perimeter, and $H$ is a half space. Then $P(F\cap H)\le P(F)$. In words: restricting a Caccioppoli set to a half-space will not increase the perimeter.
My ...
- 320
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Questions about shear transformations
I am interested in the following shear transformation $T$, which is the linear transformation on $\mathbb{R}^n$ such that the $n$ by $n$ matrix representation is given by $T = I_n + ce_n e_1^{\perp}$ ...
- 83
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87
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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
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63
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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
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56
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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 ...
- 481
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96
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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 ...
- 441
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67
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Limits of branched minimal immersions into the sphere
Can a sequence of branched minimal immersions $M_j^n$ in the round sphere $S^{n+1}$ converge to a smoothly embedded $\Sigma$, in the sense that $ M_j \to 2 \Sigma$ as currents or varifolds?
The case ...
- 5,048
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87
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Symmetry of the isoperimetric profile
Given a probability measure $\mu$ on a metric space $(X, \mathsf{d})$, the $(\mu-)$Minkowski content of a set $A$ is defined as
$$\mu^+ (A) := {\lim\inf}_{r \to 0^+} \frac{\mu ( A_r \setminus A)}{r},$$...
- 801
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126
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Absolute continuity of the volume growth in a metric space
Let $(M,d)$ be a metric space (separable, complete, better?) and let $\mu$ be a ($\sigma$-additive, positive, locally finite, regular?) Borel measure on $M$. For $x\in M$ consider the volume growth ...
- 1,959
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60
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Cardinality of intersections of lines with irregular 1-sets in the plane
From Falconer's book (The geometry of fractal sets), Lemma 3.2 says that the intersection of irregular 1-sets with straight lines is of zero $H^1$ measure. What do we know about the cardinality of ...
- 11
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0
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102
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Plateau problem in the disk: a question about geodesic nets
Consider given a finite collection of points along the boundary of the unit disk $D \subset \mathbf{R}^2$:
\begin{equation}
p_1,\dots,p_{2n} \in \partial D.
\end{equation}
We assume that these are all ...
- 5,048
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0
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97
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Reference request: theory for local minimizers in the calculus of variations
Let $F: \mathbb{R}^n \times \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}$ be the Lagrangian. We say that $f \in X$ is a local minimizer of the variational integral if for all compact sets $C \subset \...
- 161
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40
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Singular asymptotic limits of mean-convex MCF
Let $(M_t \mid t \geq 0)$ be a mean-convex mean curvature flow of hypersurfaces in ambient Riemannian manifold $(N^{n+1},g)$. Brian White proved that this flow (defined 'weakly' as a level set flow ...
- 5,048
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0
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57
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Uniform bound on the measure of $\Omega_\delta = \Omega \cap \delta\mathbb Z^d$ if $\Omega$ is an open bounded set with Lipschitz boundary
Let $\Omega \subset \mathbb R^d$ be an open bounded set with Lipschitz boundary. Let us consider $\Omega_\delta = \Omega \cap \delta\mathbb Z^d$ for $\delta >0$. I want to say that the measure of $\...
- 131
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0
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75
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Reference for calibration method for minimization problems
I am currently studying GMT and a topic that has popped up in the course is the use of calibrations as a tool for proving that a particular set $E$ attains the minimum for the problem
$$min \left \...
- 697
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0
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98
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Measure estimates of $\delta$-neighbourhood of compact sets
I am interested in the estimating from above the measure of a compact set $K$ by a sequence of sets $K_n$, converging to it in the Hausdorff metric. As such I am looking for known conditions that give ...
- 633