Questions about geometric properties of sets using measure theoretic techniques; rectifiability of sets and measures, currents, Plateau problem, isoperimetric inequality and related topics.

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33 views

C^1 Sard related question

Let $X$ be a $k+1$ rectifiable set with finite $k+1$ Hausdorff measure in $\mathbb{R}^{n+1}$ and set $Z=\{x\in X;\;\ \textrm{s.t} \;\; e_{n+1}\perp T_xX \}$, where $T_xX$ is the approximate tangent ...
4
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0answers
194 views

Area defined with $\pm$ closedness

Denote $B_n\subset\Bbb R^n$ to be unit ball at origin. Denote $S\subset B_n$ to region of type $\mathsf I$ if it satisfies $$s\in S\iff\forall t\in S, s+t\in S\mbox{ or }s-t\in S$$ I am convinced ...
1
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1answer
54 views

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 ...
5
votes
1answer
246 views

Transcendental distance sets

Define a set $S \subset \mathbb{R}^d$ as a transcendental distance set if the distance between any pair of distinct points of $S$ is transcendental. For example, $S = \{ k \, \pi \;\mid\; k=1,2,\ldots ...
2
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2answers
146 views

What can we say about the boundary of the level set of a Sobolev function?

I'm a beginner of the area of free boundary problem. Let me first give some background: $\Omega \subset \mathbb{R}^n$ is an open connected set, and locally $\partial \Omega$ is a Lipschitz graph. ...
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0answers
150 views

What does the following space look like?

Pick fixed $a=(a_1,a_2,\dots,a_d)\in\{\pm1\}^d$. Consider map $F_a:\underbrace{\Bbb R^n\times\dots\times \Bbb R^n}_d\rightarrow\Bbb R^n$ given by $F(x_1,\dots,x_d)=\sum_{i=1}^da_ix_i$. Denote ...
2
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0answers
73 views

What dimension bound is known on the singular set of a linear combination of eigenfunctions of Laplacian?

Let $(M,g)$ be a smooth, closed Riemannian manifold and suppose that $\phi_1,\dots,\phi_m$ are eigenfunctions of the Laplacian on $M$. Write $f = \phi_1 + \dots + \phi_m$. How big can the set ...
5
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0answers
45 views

Measure-minimizing simplex with fixed inradius

Let $\Delta^n$ be an $n$-simplex in $\mathbb{R}^n$. Let $V$ be the volume and $r$ the inradius (radius of the inscribed sphere) of $\Delta^n$. There is a well-known result that $$ V \geq ...
0
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0answers
59 views

Distortion of the Hausdorff dimension of sums of Cantor sets under local scaling

The following question deals with possible distortion of the Hausdorff dimension of sums of Cantor sets as one "zooms in" on the sum around any given point. Let us assume that $C_1$ and $C_2$ are two ...
8
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2answers
197 views

Continuity of length and area

Let $C_n$ be a sequence of rectifiable simple closed curves in $\mathbb{R}^2$ that converge to a rectifiable simple closed curve $D$ in the Hausdorff topology. It is easy to construct examples where ...
2
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1answer
56 views

Convex hulls have longer boundaries

Let $C$ be a rectifiable simple closed curve in $\mathbb{R}^2$ and let $D$ be the boundary of the convex hull of the region bounded by $C$. What is the most efficient way to prove that $D$ is ...
0
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3answers
169 views

When Banach indicatrix is measurable?

Let $f:X\to Y$ is a measurable function. Banach indicatrix $$ N(y,f) = \#\{x\in X \mid f(x) = y\} $$ is the number of the pre-images of $y$ under $f$. If there are infinitely many pre-images then ...
8
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1answer
151 views

Best Hölder exponents of surjective maps from the unit square to the unit cube

The Peano's square-filling curve $p:I\to I^2$ turn's out to be Hölder continuous with exponent $1/2$ on the unit interval $I$ (a quick way to see it, is to note that $p$ is a fixed point of a ...
1
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1answer
143 views

Singularities in minimal surfaces [closed]

There is a theorem by Jean Taylor that says that an almost minimal set in $\mathbb{R}^3$ can be locally parametrize by the only three possible minimal cones in $\mathbb{R}^3$, the plane, an $Y$ times ...
0
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1answer
117 views

Probability Content of a random ball in R^n

As a follow up to this question, concerning this paper: Given random variables $X_1,\ldots,X_N,X_q:\Omega\rightarrow\mathbb{R}^d$, where $X_1,\ldots,X_N$ are independent and identical distributed. ...
4
votes
1answer
190 views

Hausdorff measure of the graph

Is there any example of a real valued function on the real line whose domain has Lebesgue measure zero but the graph (in the plane) has positive one dimensional Hausdorff measure? Of course if such ...
5
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3answers
255 views

How can dimension depend on the point?

Let $M$ be a metric space. For any subset $A\subset M$ let $\dim(A)$ denote its Hausdorff dimension. For $x\in M$, define the dimension of $M$ at $x$ by $\dim(x)=\lim_{r\to0}\dim(B(x,r))$; this limit ...
0
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0answers
62 views

Derivative of Log_p map

If $\mathscr{M}$ is a $(d-dimensional)$ Riemann Manifold and $p$ is a point therein. What is the derivative of $Log_p$ function?
2
votes
2answers
80 views

understanding the average height of a unit hyper-semisphere

According to my calculations, the average height of the unit hyper-hemisphere in $\mathbb{R}^n$ is given by the formula $$h_n:=\frac{S_{n-2}}{V_{n-2}} \int_{r=0}^1 r^{n-2} \sqrt{1-r^2} dr$$ where ...
2
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1answer
87 views

How often can a single length occur as a boundary distance?

Given a bounded domain $\Omega\subset\mathbb R^n$ ($n\geq2$), how often can a single real number $r>0$ appear as a distance of two points on $\partial\Omega$? We can make any assumptions about the ...
11
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2answers
535 views

What is the Hausdorff dimension of this fractal?

Let $\sum_{i=h}^\infty d_i/b^i $ be the base $b$ representation of $x \geq 0,$ where $b>1$ and the $d_i$ are uniquely determined by the greedy algorithm. For fixed $c>1,$ let $f(x)= ...
3
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1answer
49 views

Subsets of sets of positive Hausdorff dimension with controlled upper Minkowski dimension

Call a Borel set $A \subseteq [0,1]$ good if $$0 < \dim(A) \leq \overline{\dim_\text{M}}(A) < 2 \dim(A),$$ where $\dim(A)$ is the Hausdorff dimension of $A$ and $\overline{\dim_\text{M}}(A)$ is ...
1
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1answer
75 views

“Schwarz symmetrization” on annulus

If $\Omega=\{x\in \mathbb R^n| 0<r_0<|x|<r_1\}$ is an annulus on $\mathbb R^n$, I am looking for a symmetrization result on $\Omega$. To be precise, for any $u \in W_0^{1,2}(\Omega)$, can we ...
1
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0answers
74 views

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 ...
0
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0answers
73 views

Must the Lebesgue measure of a $\rho$ - neighbourhood of an $(n-2)$ - dimensional set be at least $c\rho^2$?

The Lebesgue measure of a $\rho$-neighbourhood of a point in $\mathbb{R}^2$ is of course equal to $c\rho^2$. Similar such considerations in higher dimensions lead me to the following question: Given ...
7
votes
3answers
208 views

Set with small internal radius, small perimeter and prescribed area

Given a regular set $E\subset \mathbb R^2$ define $$ R(E) = \sup\{r\colon \exists x,\ B(x,r)\subseteq E\} $$ to be the radius of the largest circle contained in $E$ and let $|\partial E|$ be the ...
1
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0answers
95 views

Does Newtonian capacity increase strictly when mass is spread?

We start with two disjoint compact sets A and B with positive capacities. Then, we translate B s.t. $B+rv$ is disjoint from A and B and ,more importantly, $dist(x,y)<dist(x,y+rv)$ for all $x\in A$ ...
4
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1answer
142 views

Calculate Hausdorff measure with Frostman measures

Fix a metrix space $(X,d)$ and consider the Hausdorff (outer) measures $\mathcal{H}^s$ on $X$. A Frostman measure on $X$ is a finite Borel measure $\mu$ such that there exists $C,t,r_0>0$ with ...
3
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1answer
216 views

The relation between Hausdorff dimension of an $n$-manifold and $n$

It is known that for a topological space with different metrics, the Hausdorff dimensions may not be equal in general. For the case of manifolds, suppose $M$ is a $n$-manifold with a ...
2
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0answers
47 views

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 ...
0
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0answers
46 views

Is the blowing up the rectifiable set cone?

Let $M$ be a rectifiable set in $\mathbb{R}^N$. For any point $p\in M$, is the following true?: $\lambda_i M$ subconverges to a cone in $p$ for $\lambda_i\to\infty$, i.e. $(\lambda_i M)\cap B(p,R)$ ...
1
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1answer
54 views

Convergent algorithm for dividing a body into two regions of equal volume

Let $\Omega \subset R^3$ be a bounded open region. It is well known that there exists a smooth surface $\Gamma$ with minimum area and constant mean curvature which is orthogonal to $\partial \Omega$ ...
2
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0answers
88 views

Besicovitch's covering theorem for ellipsoids and shadows

The usual Besicovitch's covering theorem concerns closed balls in $\mathbb{R}^d$. It relies on a property called "directionally limited metric space": the principal ingredient is to say that there ...
5
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1answer
143 views

Averaging maps of Riemannian manifolds

Let $M$ be a compact Riemannian manifold. We know how to average functions $f\colon M\to {\mathbb R}$; the integral $\frac{\int_M f}{\int_M 1}$ returns a value in ${\mathbb R}$. If intead $f\colon ...
6
votes
1answer
232 views

Precise density estimates for Cantor sets

Let $C_\lambda$ be the classical Cantor set associated to a real number $0<\lambda<\frac{1}{2}$, as defined for example in the book of K. J. Falconer The geometry of fractal sets. I recall ...
5
votes
2answers
216 views

Stability of minimal surfaces

Let $\Gamma$ be a prescribed $n-2$ dimensional set and assume $S \subset R^n$ is a minimal hyper-surface with respect to some smooth metric $g$ on $R^n$, and $\partial S= \Gamma$. Is $S$ is stable ...
0
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0answers
77 views

existence of locally translation-invariant Borel measure on Frechet manifolds

It is well known that the only locally finite, translation-invariant Borel measure on an infinite-dimensional, separable Frechet space is the trivial measure. I am wondering about an analogous ...
3
votes
0answers
143 views

Hausdorff densities

I've been stuck on this one for a while now. Given an $\mathcal{H}^{s}$ measurable subset $E\subset \mathbb{R}^n$ with $0<\mathcal{H}^{s}(E)<\infty$, we let $\overline{D}^{s}(E,x)$ denote the ...
0
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1answer
178 views

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
vote
1answer
96 views

Is it true that $ H^{n-1} (spt \mu _E - \partial ^{*}E)=0 ?$

In Federer's Theorem, $ H^{n-1} (\partial ^{m}E - \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 $\partial ^{*}E$ ...
12
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1answer
497 views

Open problems in Federer's Geometric Measure Theory

I wanted to know if the problems mentionned in this book are solved. More specifically, at some places, the author says that he doesn't know the answer, for example :"I do not know whether this ...
3
votes
1answer
130 views

Wiener measure of hitting sets A,B but not C (or easier hitting A but not C)

I am trying to formulate the measure of event $E=\{B[0\infty)\cap A,B \neq \varnothing$ and $B[0\infty)\cap C= \varnothing\}$, where $B[0\infty)$ is a Brownian path and $A,B,C$ are pairwise ...
6
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1answer
324 views

Geometric measures different from Hausdorff

$\newcommand{\RR}{\mathbb{R}}\newcommand{\calF}{\mathcal{F}}\newcommand{\diam}{\mathrm{diam}}$ In geometric measure theory there are various notions of $m$-dimensional measure for sets $A\subset ...
2
votes
0answers
61 views

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 ...
3
votes
1answer
254 views

An integral estimate over rotations of the dyadic grid

I'm currently reading the paper Rectifiable Sets and the Traveling Salesman Problem (link) by Peter Jones (Invent. math. 102, 1-15 (1990)), and am having trouble understanding an integral estimate ...
1
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1answer
124 views

Regularity of intersection of Lipschitz boundaries

I am interested in the regularity of the "$n-1$ dimensional boundary" of the intersection of two Lipschitz boundaries, in particular I would like to know if this boundary always has zero $n-1$ ...
2
votes
0answers
79 views

isoperimetric problems on Alexandrov spaces

For an Alexandrov space M with curvature bounded from below, the isoperimetric profile $v \to I_M(v)$ defined for every $v\in (0,V(M))$ (the volume of M might be infinite), is given by $$ ...
8
votes
1answer
296 views

Besicovitch Covering Lemma on Manifolds

The classical Besicovitch covering lemma (BCL) asserts that for any $d \geq 1$, there is a constant $N(d)$ with the following property. If $A \subset \mathbb{R}^d$ is any subset and $r : A \to (0,R]$ ...
2
votes
1answer
130 views

The Mahler conjecture and non-zonoidal 3-polytopes (4-polytopes)

I have been working on the Mahler conjecture for over a year now and have made some progress for certain classes of convex polytopes and I'm now attempting to write up my results specified to ...
2
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0answers
151 views

Convex functions with non-singular hessian measure are continuously differentiable?

It is known that every convex function $f: \Omega\to \mathbb{R}$, $\Omega$ convex subset of $\mathbb{R}^n$, has a weak derivative of bounded variation $Df\in BV_{loc}(\mathbb{R}^n)$ (e.g. Evans and ...