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.
287 questions with no upvoted or accepted answers
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About the current of finite mass
In Demailly's e-book Complex analytic and differential geometry,
chap3-(1.14) Proposition is stated as follows:
Every positive current $T=i^{(n-p)^{2}} \sum T_{I, J} d z_{I} \wedge d \bar{z}_{J}$ ...
- 391
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126
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Is Steiner symmetrization "Turing complete"?
This question stems from intuition so it is a little soft. It concerns performing computation using transformations on sets. The idea is that a rearrangement like Steiner symmetrization might be "...
- 203
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0
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92
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First Dirichlet eigenvalue below second Neumann eigenvalue?
Let $\Omega$ be a bounded domain in $\mathbb R^n $ with smooth boundary.
I was wondering if there exist any known conditions on $\Omega$ such that the 1st Dirichlet eigenvalue of the (positive) ...
- 173
2
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0
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178
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Caratheodory's theorem in any compact Riemann surface
The classical Caratheodory theorem states that if $G$ is a simply connected domain in the plane, whose boundary is a Jordan curve, then the Riemann uniformization extends continuously to a ...
- 377
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90
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Obstacle problems for minimal hypersurfaces
Given a compact Riemannian $n+1$-manifold $M$ with (possibly not mean convex) boundary (smooth or probably with codim $>2$ singularities). Consider the following problems,
1) fix a homology class $...
- 123
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92
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lower volume bound of submanifolds with small mean curvature integral data
$(M^n,g)$ is a smooth submanifold in $\mathbb{R}^p$ ,and $B_1$ is the unit ball centered in the origin 0.
Is there a $\epsilon >0$，
when assuming $\int_{M\cap B_1} |H|^n \leq \epsilon$, and the ...
- 101
2
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0
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90
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Invariance under diffeomorphisms of the Hajlasz-Sobolev spaces
In this post it was shown that if $\Omega$ and $\Omega'$ are diffeomorphic non-empty open domains in some Euclidean space then the corresponding local Sobolev spaces are diffeomorphic with ...
- 5,405
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248
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Intrinsic volume - is there a simplified formula?
I'm struggling to understand how can I compute the instrinsic volumes (or Minkowski functionals) of a submanifold $\Omega$ of $\mathbb{R}^N$. I found a formula, but I really can't understand it...
It ...
- 1,759
2
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71
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Is there a function $u \in BV_{loc}(\mathbb{R}^2)$ such that $Du$ is an $s$-dimensional Hausdorff measure restricted to the Koch curve?
Motivated by my previous question Alberti rank-one theorem and irregular jump discontinuities, I'd like to ask the following:
Is there a function $u \in BV_{loc}(\mathbb{R}^2)$ such that $Du$ is an ...
- 839
2
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73
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Alberti rank-one theorem and irregular jump discontinuities
Is it fair to say that Alberti rank one theorem means that a BV functions $u \in BV(\mathbb{R}^2)$ has $Du = D^{cantor}u$ if and only if it has a jump discontinuity across a curve that is not smooth (...
- 839
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73
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Projection of BV function
Let $u \in [BV(\mathbb R^N)]^N$. We have
$$D^{jump} u(x) = a(x) \otimes b(x)|D^{jump}u|,$$
where $a,b \in \mathbb S^{N-1}$.
What is the projection of $D^{jump}u$ in the direction $a$? And how can ...
- 839
2
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199
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Convergence of the difference quotient of a BV function
Consider a BV function $u \in BV(\mathbb{R}^N; \mathbb{R}^N)$.
What can be said about the difference quotient
$$
\frac{u(x+\epsilon y)-u(x)}{\epsilon}
$$
regarding its convergence as $\epsilon \to 0$...
- 839
2
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164
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Jacobian and Jacobian matrix of solutions of ODE with Sobolev vector field
Let $\Phi$ be the Lagrangian flow (defined as in page 6 of this paper) of the ODE
$$\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{...
user124345
2
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279
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Relationship between $p$-capacity and Riesz $s$-capacity of a set
What is the relationship between the definitions of $s$-capacity (page 13 here) and $p$-capacity (here) of a set?
Are they equivalent? If not, what inequalities hold? What is the difference (in terms ...
- 839
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Role of absolute continuity of divergence of BV function in proof of renormalization property
In the paper http://cvgmt.sns.it/paper/436/, the author proves the renormalization property for the flow generated by a vector field $a(t,\cdot) \in BV(\mathbb{R}^N; \mathbb{R}^N)$.
Heuristically, ...
- 839
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92
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Lower bound to $\epsilon$-expansion of a subset of a half-sphere
Below are two known lemmas on a $d$-dimensional sphere (related to the isoperimetric inequality). I would like to know: does a similar statement like this holds for a $d$-dimensional dome also (i.e. ...
- 21
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159
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What is the boundary of the set $\{ x : dist (x ,\partial \Omega) > \alpha \}$ for a domain $\Omega$?
Let $\Omega$ is a bounded open domain in $\mathbb R ^n$, and $\alpha \geq 0$ a real number, and consider the set $ E_\alpha = \{ x \in \Omega : \text{dist}(x , \partial \Omega) > \alpha\} $, which ...
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Is a maximal set of rectangles known for which Lebesgue’s Differentiation Theorem holds true?
Lebesgue's differentiation theorem states that if $x$ is a point in $\mathbb{R}^n$ and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a Lebesgue integrable function, then the limit of $\frac{\int_B f d\...
- 4,597
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Volume of critical points decreases under symmetric decreasing rearrangement
In the lecture note http://www.math.utoronto.ca/almut/rearrange.pdf, it was stated that the volume of the set of critical points decreases under symmetric decreasing rearrangement. It seems so obvious ...
- 21
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Lebesgue density theorem for "doubling uniformly covering collections of subsets"
I am looking for a version of Lebesgue density theorem that works when restricting to "good" collections of balls with respect to (not necessarily doubling) metric measure spaces. Specifically
Let $(...
- 2,964
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115
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about the compactness of minimal surfaces
If a Caccioppoli set $A$ is of minimal perimeter in every compact set $K$ contained in some open set, can we say that $A$ is of minimal perimeter in the open set? If not, please construct a ...
- 181
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98
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Points on Sphere whose image, under symmetric positive definite matrix, is contained in cube
Let $\Sigma \in \mathbb{R}^{n \times n}$ be a symmetric, positive definite matrix and let $\mu_r$ denote surface measure on the sphere in $\mathbb{R}^n$ with radius $r$. Let
$$
R = \{x \in \mathbb{R}^...
- 143
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68
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Are these two sets always coincide after a translation or scaling?
I get stuck with the following problem, which I think is related to sum-product estimate. Here is the problem.
Problem
Given two sets $A, B\subset \mathbb R^n$, and a sires of positive number $\...
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78
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A uniform version of Minkowski content?
Let $A\subseteq [-1,1]^d$ be a measurable set and $\mu$ be the Lebesgue measure. For any $\delta>0$, define $A_\delta := \{x: d(x,A)\leq \delta\}$, where $d(x,A) := \inf_{y\in A}\|x-y\|_2$.
The ...
- 373
2
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0
answers
899
views
norm of projection of a random vector on the sphere onto a linear subspace
Let $x$ be a random vector uniformly distributed on the unit sphere $\mathbb{S}^{D-1}$ and let $\mathcal{V}$ be a linear subspace of dimension $m$. Then it is known that the euclidean norm of the ...
- 398
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108
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Quantitative estimates on space filling curves
To my understanding, quantitative topology/geometry makes statements quantitative. Examples: 1. a quantitative version of Invariance of Dimension is waist inequality. 2. Lusternik-Fet says a closed ...
- 365
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99
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Relationship between weight of spanning tree in a tree metric approximation and the original metric
So suppose we have a tree metric which approximates the Euclidean distance between a finite set of points. The leaves correspond to points in the original space. It may be an ultra metric, and ...
- 161
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79
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Compute Mixed Volume with Respect to Some Regular Sets
Let $( \mathbb{R}^n, \mathcal{B}, \gamma)$ be a measure space where $\mathcal{B}$ is the Borel sigma algebra and $\gamma$ is a continuous measure. For $A, B\in \mathcal{B}$ that are convex, the mixed ...
- 1,127
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115
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Currents with mean curvature
so let $T=\tau(M,\theta,\xi)$ a rectifiable current in $U\subset\mathbb{R}^{n+k}$, then $T$ has mean curvature vector $H$ if for every $X\in C^1_c(U\setminus\partial T,\mathbb{R}^{n+k})$ the variation ...
- 43
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285
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Method of proving the regularity of the minimizer of geometric variational problems
Recall the proof of the $(\Lambda,r_0)$-perimeter minimizer.
We say that $E$ is a $(\Lambda,r_0)$-perimeter minimizer in an open set A provided $sptD\chi_E=\partial E$ and there exist two constants $...
- 1,350
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142
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sets with positive reach with complementary set with positive reach
I am interested in bibliographical references about a special class of sets, those who have positive reach and which complementary has also positive reach.
I recall that the reach $R\geq 0$ of a set ...
- 1,352
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139
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Probability of hitting a Borel set by transient Brownian motion ($d\geq 3$)
I am looking for references/progress made in estimating the hitting probability for Borel sets.
For spheres we have $P_{x}(T_{B_{r}(0)}<\infty)=(\frac{|r|}{|x|})^{d-2}$, where $x=B_{0}$ for ...
- 5,474
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answers
101
views
Measure of points with small neighborhood in convex bodies
Let $K \subseteq \mathbb{R}^n$ be a "fat" convex body, i.e. one that contains a ball of radius 1. I'm interested in the following question about points $y \in K$: If you take a normally distributed $e ...
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212
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Can a compact metrizable space be determined by its Hausdorff measures?
Suppose that $(X,d)$ is a compact metric space. Now suppose that $h:[0,a]\rightarrow[0,b]$ is a continuous function with $h(0)=0$ where if $x\leq y$, then $h(x)\leq h(y)$. Then define $$L(d,h)=\lim_{\...
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201
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Bonnesen's inequality for non-simple curves
Given a closed curve in the plane $\mathbb{R}^2$, it is well known that $L^2 \geq 4\pi A$ where $L$ is the length of the curve and $A$ is the area of the interior of the curve.
For a simple closed ...
- 2,641
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175
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An isoperimetric type maximization problem with a barrier.
I'm trying to minmize a particular functional which depends on a curve with fixed endpoints which lies below a fixed line in $\mathbb{R}^2$. Here are the details:
Let $(r(\theta), \theta)$ be a ...
- 2,641
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108
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candidates for "projection" of the trace of a set onto the associated perimeter minimizer
Let $E$ have finite perimeter in $\mathbb R^n$. Consider the minimization of
$$P(B)-\int_{\mathbb R^n\setminus L}|D\chi_E|$$
among the sets $B$ of finite perimeter differing from $E$ only inside $L$ ...
- 2,041
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1
answer
624
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Results for Hausdorff Measure after Linear Transformation
For the Sierpinski Triangle, $S$, the $d$ dimensional Hausdorff measure is given by, $H^{d}(S)$. If a linear transformation, $W$ is applied to $S$, with
$$W(x,y)=\begin{bmatrix} 1/2 & 0 \\ 0 &...
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0
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51
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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 $T = I_n + ce_n e_1^{\perp}$ ...
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0
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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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0
answers
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
1
vote
0
answers
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 ...
- 5,038
1
vote
0
answers
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
1
vote
0
answers
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
1
vote
0
answers
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
1
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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 ...
- 5,038
1
vote
0
answers
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
1
vote
0
answers
40
views
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 ...
- 5,038