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2
votes
2answers
126 views

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 ...
3
votes
1answer
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.
9
votes
3answers
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 ...
1
vote
1answer
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 ...
1
vote
0answers
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 ...
5
votes
2answers
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 : ...
2
votes
0answers
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 ...
1
vote
0answers
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 ...
1
vote
1answer
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 ...
0
votes
0answers
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 ...
9
votes
1answer
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 ...
2
votes
1answer
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 ...
2
votes
1answer
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 ...
0
votes
1answer
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 ...
0
votes
2answers
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 $n-1$?
0
votes
0answers
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 ...
2
votes
1answer
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 ...
1
vote
0answers
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 ...
2
votes
0answers
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 ...
2
votes
1answer
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 ...
3
votes
1answer
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, ...
1
vote
1answer
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 $. ...
1
vote
1answer
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 ...
2
votes
1answer
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 ...
1
vote
0answers
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 ...
1
vote
1answer
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 ...
1
vote
2answers
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 $BV\cap L^\infty$.
7
votes
2answers
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 ...
2
votes
0answers
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 ...
4
votes
1answer
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 ...
4
votes
0answers
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 ...
2
votes
3answers
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 ...
3
votes
1answer
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 ...
0
votes
0answers
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 ...
7
votes
1answer
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 ...
0
votes
0answers
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 ...
15
votes
4answers
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 ...
13
votes
0answers
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 ...
5
votes
1answer
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 ...
2
votes
1answer
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” ...
0
votes
1answer
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. ...
2
votes
1answer
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 ...
0
votes
1answer
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 ...
1
vote
0answers
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 ...
14
votes
0answers
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 ...
5
votes
1answer
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 ...
2
votes
1answer
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 ...
1
vote
0answers
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 ...
5
votes
0answers
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 ...
2
votes
0answers
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 ...