The geometric-measure-theory tag has no wiki summary.

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### Perimeter of a 'trapped' convex set

Consider the following setup: three bounded, 'nice' convex sets $A \subseteq B \subsetneq C \subset \mathbb{R}^2$, and three points $x,y,z\in \partial A\cap \partial B\cap \partial C$ (see edit ...

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**1**answer

69 views

### Is the countable intersection of residual sets in [0,1] with Hausdorff dimension 1 of full Hausdorff dimension?

Let $E_k\subset [0,1]$ be residual subsets (i.e. containing dense $G_\delta $ set) with $E_{k+1}\subset E_k$ and $\dim_HE_k=1, \forall k.$ My question is : $\dim_H\bigcap_k E_k=1?$ Thanks.

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456 views

### Does the centroid depend continuously on the curve?

Let $\gamma$ be a piecewise smooth curve in $\mathbb{R}^n$. Recall that the centroid of $\gamma$ is the point $(\overline{x}, \overline{y})$ where $\overline{x}$ is the average value of $x$ on ...

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**1**answer

85 views

### Two equivalent measures on the real Grassmannians?

Denote by $G(n,k)$ the real Grassmannian, the set of $k$-dimensional subspaces of $\mathbb{R}^n$. It is a topological space, even metrizable (see A metric for Grassmannians), and so it is a ...

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74 views

### Finding density of Haar measure relative to the Liouville volume measure on $\frac{G^{\mathbb{C}}}{P}$

Let $G$ be a connected compact Lie group and $G^{\mathbb{C}}$ be the complexification of the Lie group $G$ then we know that by polar decomposition we can write $G^{\mathbb{C}}\cong G\times ...

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206 views

### Can Hausdorff dimension make sets into a Tropical Semiring?

If $X$ is a metric space, we construct Hausdorff $d$ measure from the outer measure
\begin{equation}
H^d(U) = \lim_{\delta \to 0}\inf\left\{\sum_{i=1}^\infty \left(\text{diam}(E_i)\right)^d : ...

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44 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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149 views

### Brakke's theorem for gap in entropy between self-shrinkers

In their paper The round sphere minimizes entropy among closed self shrinkers, Colding-Ilmanen-Minicozzi-White state "It follows from Brakke's theorem that $\mathbb{R}^n$ has the least entropy of any ...

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**1**answer

129 views

### 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 ...

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88 views

### Convergence of functions of bounded variation

Let $u_n$ be a sequence of smooth functions with $||u_n||_{W^{1,1}}<C<\infty$. By $BV$ compactness, a subsequence of $u_n$ converges to some $u$ in $L^{n/(n-1)}$. Assume $\int_{\Omega} (|\nabla ...

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**1**answer

204 views

### Lower-Hölder embeddings of the sphere

My question is very simple:
Given $d\ge 3$, does there exist $s\in (0,1)$ and an embedding $f:S^{d-1}\to \mathbb{R}^d$ such that
$$
|f(x)-f(y)| \ge |x-y|^s \quad\textrm{if } |x-y|<r,
$$
for ...

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**1**answer

98 views

### Compact Riemannian manifold with maximum average distance

Given a compact connected Riemannian $2$-manifold $M$ with positive curvature (thus by Gauß-Bonnet, $M$ is diffeomorphic to a 2-sphere) and diameter 1, what is the supremum (as $M$ varies over all ...

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**1**answer

117 views

### Function from a compact metric space to the subsets of the naturals

Let $X$ be a compact metric space, and $\mu$ a Borel probability measure. For
$S\subset\mathbb{N}$ we denote the upper density with $\overline{D}(S).$
Let $f:X\rightarrow2^{\mathbb{N}}$ be a ...

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**1**answer

99 views

### Lipschitz boundary vs rectifiable curve boundary

I was looking at an old paper about domains with Lipschitz boundary. I am wondering, suppose that the boundary of a compact domain homeomorphic to a disk is a rectifiable injective curve : is this ...

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80 views

### Hausdorff measure of the zero set

Let $f : \mathbb R^n\to \mathbb R$ continuous, for which there exist $x,y\in\mathbb R^n$, such that $f(x)f(y)<0$.
Is it true that the Hausdorff dimension of the zero set of $f$ is at least $n-1$?

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48 views

### Minimal length of connection

I wonder if someone has an example which answers the following problem.
Find $X\subseteq\mathbb R^n$ bounded, such that the set
$$
\{ \mathcal H^1(S) \colon S\cup X \text{ connected}\}
$$
does not ...

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**1**answer

156 views

### A special case of the Divergence theorem

I am interested in the following statement:
Let $F$ be a vector field in $\mathbb{R}^n$ that is $C^1$-smooth in a
domain $U$, continuous up to the boundary $\partial U$, and vanishing on ...

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787 views

### Are planar Lipschitz curves countable unions of graphs?

More precisely:
Question:
Let $\gamma \colon [0,1] \to \mathbb{R}^2$ be Lipschitz. Do there exist Borel (or Suslin) sets $A_i \subset \mathbb{R}^2$ and directions $v_i \in \mathbb{R}^2$, for ...

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71 views

### The constant vector field lemma in White's paper [closed]

When I read White's paper (Comment. Math. Helvetici, 1989) named: "A new proof of the compactness theorem for integral currents", I am confused about lemma 2.2, the constant vector field lemma. Can ...

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**1**answer

124 views

### Are there any good techniques for calculating Hausdorff measure?

I'm aware that many techniques have been developed for the purpose of calculating Hausdorff dimension (although I'm fairly unfamiliar with them), but my question is whether or not we have any good ...

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**1**answer

230 views

### How to define a generalized differential form through its values on submanifolds

Suppose we're in $\mathbb R^n$, and we have a function on line segments ,$\omega(I)$, with values in $\mathbb R$. Give sufficient conditions for $\omega$ to be given by a generalized 1-form (that is, ...

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107 views

### Hausdorff measure and projections

Fix $ k \in \mathbb{N} $ and let $ H^k $ be the $k$-dimensional Hausdorff measure on $\ell^\infty $. Also, if $ V $ is a subspace of $ \ell^\infty $, we denote the projection onto $ V $ by $ \pi_V $. ...

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213 views

### (n-1)-dimensional normal currents and Smirnov's paper

I don't know much about currents, but I saw a paper of Smirnov which seems relevant to a problem I am working on. In the very last paragraph of page 848 of the following paper
...

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155 views

### $X$ Polish geodesic implies $(P_2(X), W_2)$ geodesic

If $X,d$ is a complete and separable space then the space of Borel probability measures with finite second moment on $X$ endowed with the Wasserstein distance $W_2$ is geodesic.
I am looking for ...

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131 views

### 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 ...

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93 views

### Monotonicity and perturbation of $J$-holomorphic curves

In a symplectic manifold $(M, \omega)$, $J$-holomorphic curves are special minimal surfaces if $J$ is compatible with the symplectic structure and the metric is induced by $\omega$ and $J$.
Minimal ...

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216 views

### Banach algebra of BV functions

I would like to find a reference for the proof that functions of bounded variation make a Banach algebra. Same question for $BV\cap L^\infty$.

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273 views

### Measures whose projections are absolutely continuous

Since my question was not answered on MSE, I would like to ask it here.
Let $\mu$ be a finite Borel measure on the plane. Does there exist a characterization of the property that almost all (wrt ...

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64 views

### Do the interiors of curves of constant width admit “linear” measures?

Concise question: In two dimensions, do all shapes of constant width admit a measure over their interior such that for any two parallel lines intersecting the shape, the area between them under the ...

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**1**answer

90 views

### dimension of a union of grassmannians

I'm working on a dynamical systems problem and have arrived at a naive question in differential topology? geometric measure theory?
I have a smooth path $\gamma\colon \mathbb R\to\mathcal ...

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163 views

### decreasing rearrangements: why the asymmetry of measure-preserving maps?

Ryff proved in 1970 that the decreasing rearrangement $f^*$ of a, say, continuous function $f:[0,1]\to\mathbb{R}$ admits a measure preserving map $\phi$ such that $f=f^*\circ\phi$. In general it is ...

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411 views

### Hausdorff measure on the sphere is well defined?

Given $n\in\mathbb{N}$, consider the $\ell_2$ unit sphere $\mathbb{S}^{n}\subset\mathbb{R}^{n+1}$ equipped with its "geodesic" metric $\rho_n$ defined as:
$\rho_n(x,y)=\arccos \Big(\langle ...

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200 views

### Technical question on perimeter of level sets

Sorry for asking such a basic question, but this is not my area of expertise.
In my work I'm using the coarea formula: for $\Omega \subseteq \mathbb{R}^n$ open and bounded, and $u : \Omega \to ...

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53 views

### Approximation of sets from the interior such that perimeters converge

Let $A\subset \mathbb{R}^n$ be an open bounded set that is the interior of a closed set. Does there exist a sequence of open sets $A_j\subset \subset A$ such that the (Lebesgue) measure $|A\setminus ...

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239 views

### Doubling space without Besicovitch covering theorem?

A metric space is doubling if any ball of radius $2R$ can be covered by $N$ balls of radius $R$ and $N$ is fixed once forever.
Is there an example of complete length-metric space which is ...

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161 views

### Measurable projection theorem

Hi ;
i have this theorem from the book :Set-valued analysis
Let $(\Omega,\mathcal{A},\mu)$ be a
complete $\sigma$-finite measure space
, $X$ a complete separable metric
space and ...

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860 views

### Why are currents named currents?

Why do currents, functionals on compactly supported differentiable n-forms, bear the name they do?
I've assumed that it has something to do with an electrical current being formalized as a vector ...

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190 views

### Measure Preserving Maps from the Square to the Cube

There is a measure preserving map from the unit interval onto the unit cube that is Lipschitz of order 1/2, that is $|f(x)-f(y)| \leq A |x-y|^{1/2}$. By considering the image of small intervals, one ...

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212 views

### Extension of measures from the ball sigma-algebra to the borel sigma-algebra

Let $X$ be a metric space, $\Sigma_{1}$ the borel sigma algebra and
$\Sigma_{2}$ the sigma algebra generated by balls (open and closed).
If $\mu$ is a probability measure on $\Sigma_{2}$ can it be ...

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**1**answer

179 views

### Approximating Jordan curves

I'd like to capture the intuitive notion that a Jordan curve $\gamma_2$ “follows” or “approximates” another Jordan curve $\gamma_1$, i.e. goes somehow “parallel” ...

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**1**answer

220 views

### Is there any known condition for the following property?

For a mapping $f: \Omega\to \bf{R}^n$, what kind of condition ensures that the one-dimensional Hausdorff measure of $f^{-1}(E)$ is zero whenever $E$ is of zero one-dimensional Hausdorff measure zero. ...

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281 views

### Hausdorff dimension of convex set in ${\bf R}^n$

I wanto know the smoothness of convex set in ${\bf R}^n$.
Recall the following definition.
Definition : $X$ is a bounded closed convex set in ${\bf R}^n$ if for $x$, $y\in X$, the any ...

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**1**answer

103 views

### On differentiability relative to a n-rectifiable subset of \mathbb{R^N}

Let S be a n-rectifiable subset of \mathbb{R}^N , we define the differentiability of a funtion f:S \to \mathbb{R} at a point x_0 in S as in Federer's book, where he called differentiable relative to S ...

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237 views

### Banach-Tarski vs von Neumann

While not so well known, the von Neumann paradox is built among the same lines, in dimension 2 and with transforms within the special linear group. But what is wrong with the following "proof" : it ...

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689 views

### A Kakeya-like problem: must a union of annuli fill the plane?

Let $S$ be a subset of $\mathbb{R}^2$ with the following property. For all $x \in \mathbb{R}^2$ and $\varepsilon \gt 0$, there exists a nontrivial interval $[a,b] \subseteq [1-\varepsilon,1]$, such ...

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149 views

### Minimizing the perimeter around an obstacle

Let $A\subset \mathbb{R}^n$ be a (measurable) bounded set, and consider the following optimization problem: minimize $P(X)$, the perimeter of a set $X$, where $X$ ranges over all Caccioppoli subsets ...

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225 views

### common dominating measure for a family of measures

Given a family $\{\mu \}_{i\in I}$ on a Polish space (complete, separable metric space) $X$. When does there exist a measure $\lambda$ such that
$$\mu_i=f_i \lambda$$
where the f_i are densities ...

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131 views

### 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 ...

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206 views

### Smoothness of the convolution of a singular measure with itself

Let $\gamma:I\subset\mathbb{R}\rightarrow\mathbb{R}^2, s\mapsto \gamma(s)$, denote the arclength parametrization of a smooth, convex curve $\Gamma:=\gamma(I)\subset\mathbb{R}^2$. Equip $\Gamma$ with ...

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139 views

### 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 ...