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.

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1 answer
54 views

Hahn-Mazurkiewicz with finite one-dimensional Hausdorff measure

Suppose that there is a continuous surjection from $[0,1]$ to a metric space $(X,d)$. If $(X,d)$ has finite one-dimensional Hausdorff measure, must there exist a Lipschitz surjection from $[0,1]$ to $(...
6 votes
1 answer
198 views

Densities, pseudoforms, absolute differential forms and measures, differential forms, etc

Apologies if this question is too basic, but I figured I first heard of most of these concepts on MO, so perhaps I can ask here. Gelfand’s definition, copied from AlvarezPaiva [My edit, could be ...
6 votes
0 answers
186 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 \frac{n^{n/2}(...
3 votes
1 answer
117 views

Given a set of finite perimeter $\Omega$ s.t. $\partial ^* \Omega =\partial \Omega$, it's not true that $P(\Omega)= \mathcal{H}^{n-1} (\Omega)$

In the article "Funzioni BV e tracce" by Anzellotti and Giaquinta, at page 6 you can read (assume $\Omega \subset \mathbb{R}^n$ open): "The following example shows that the hypothesis $\...
3 votes
1 answer
357 views

Local dimension of measures

For a Borel prob measure $\mu$ in $\mathbb{R}^n$, define the local dimension of $\mu$ at $x$ by $$ {\rm dim}_*(\mu, x)=\liminf_{r\to 0} \frac{\log \mu(B(x,r))}{\log r}, {\rm dim}^*(\mu, x)=\limsup_{r\...
-1 votes
0 answers
91 views

Are these two spaces of functions identical?

Let the function space $A$ denote all functions $f : [0, 1) \to [0, 1)$ such that, for some set $Z$ of Lebesgue measure zero, the derivative $f'$ exists on $[0, 1) \setminus Z$ and $|f'| = 1$ there. ...
2 votes
1 answer
102 views

On the existence of a complicated fractal-like set of finite perimeter

Let $f\in BV(\Bbb R^n)$ be an integer-valued function that maps into $\{0, 1\}$ and is identically $0$ outside some bounded set in $\Bbb R^n$. In particular, $f$ determines a bounded Caccioppoli set $...
4 votes
0 answers
68 views

Solution to the Eikonal equation with almost everywhere continuous derivative

Let $\Omega$ be an open, bounded, connected subset of $\mathbb R^n$ with smooth boundary. Does there always exist an almost everywhere solution $u \in W^{1, \infty}$ to the following system of PDE? $$|...
47 votes
4 answers
11k views

Volumes of n-balls: what is so special about n=5?

I am reposting this question from math.stackexchange where it has not yet generated an answer I had been looking for. The volume of an $n$-dimensional ball of radius $R$ is given by the classical ...
1 vote
1 answer
148 views

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)=\...
2 votes
0 answers
177 views

Statistical invariants of Riemannian manifolds

$\DeclareMathOperator\diam{diam}\DeclareMathOperator\rad{rad}\DeclareMathOperator\iso{iso}\DeclareMathOperator\com{com}\DeclareMathOperator\con{con}$A cheap way of defining invariants of Riemannian ...
0 votes
0 answers
84 views

Why does $\omega$ belong to $D^{\varepsilon^3}\theta^\ast$ for $\lesssim \rho^{-n} D^{n-2 + n\varepsilon^3}$ different $\theta$?

In this paper, there is the following claim (Pg. 1850): If $1 - \eta(w) \ne 0$, then $|\omega| \ge \rho$. In that case, $\omega$ belongs to $D^{\varepsilon^3}\theta^\ast$ for $\lesssim \rho^{-n} D^{n-...
1 vote
0 answers
79 views

Measurability of the union of cut loci along a curve

Let $(M,g)$ be a Riemannian symmetric space and $\alpha(s)$ be a geodesic. Define $$ U(t)=\cup_{s\in[0,t]}{\rm Cut}(\alpha(s)) $$ as the union of the cut loci ${\rm Cut}(\alpha(s))$ along the curve $\...
0 votes
0 answers
29 views

On the Lipschitz parametrizability of polynomials of fixed Mahler measure

Background For a polynomial $f(x) = a(x-\alpha_1) \cdots (x - \alpha_n) \in \mathbb{C}[x]$, its Mahler measure is defined to be $$M(f) = |a| \prod_{i=1}^n \max\{1, |\alpha_i|\}$$ In Lemma 1, Masser-...
2 votes
0 answers
80 views

Equivalence class of parametrized surfaces which induce the same current

Suppose $M$ is a smooth manifold of dimension $n \geq 2$. A $k$-current is a linear functional on compactly supported smooth forms on $M$, denoted $T: \Omega^k_c(M) \to \mathbb{R}$. Let $X: [0,1]^2 \...
5 votes
1 answer
380 views

Tangent cones at zero and infinity to minimal surfaces

Let $n \geq 2$, and let $M^n \subset \mathbf{R}^{n+1}$ be a minimal surface with $0 \in M$ and finite ($n$-dimensional) area growth: $\operatorname{limsup}_{R \to \infty} R^{-n} \lVert M \cap B_R \...
1 vote
0 answers
46 views

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 ...
2 votes
0 answers
137 views

$\mathscr{H}^{n-2}(\Sigma)< \infty$ implies $\mathscr{H}^{n-1}(\pi(\Sigma))=0$

Let $\Sigma\subset \mathbb{R}^{n+1}$ be a set with $(n-2)$-dimensional Hausdorff measure finite, i.e. $\mathscr{H}^{n-2}(\Sigma)<\infty$. Let $\pi:\mathbb{R}^{n+1}\to \mathbb{R}^n$ be the ...
2 votes
1 answer
266 views

If $\mathcal{H}^{n-1}(E)=0$ then $\mathbb{R}^n\setminus E$ is connected

Let $E\subset \mathbb{R}^n$ be a (measurable) subset with $\mathcal{H}^{n-1}(E)=0$, where $\mathcal H^{n - 1}$ is the ($n - 1$)-dimensional Hausdorff measure. I want to know if $\mathbb{R}^n\setminus ...
1 vote
1 answer
303 views

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 ...
0 votes
0 answers
89 views

Bounding the area of the image of a set by product of maximum of lengths

Let $F:[0,1]\times[0,1]\to \mathbb {R}^2$ be a smooth function. Given $x\in [0,1]$, let $\ell_x:=\{x\}\times [0,1]$, and given $y\in [0,1]$, let $\ell_y:=[0,1]\times \{y\}$. My question feels ...
27 votes
1 answer
1k views

The dual of $\mathrm{BV}$

$\DeclareMathOperator\BV{BV}\DeclareMathOperator\SBV{SBV}$I'm going to let $\BV := \BV(\mathbb{R}^d)$ denote the space of functions of bounded variation on $\mathbb{R}^d$. My question concerns the ...
4 votes
1 answer
171 views

What are the possible blow up limits of an $L^1$ function?

Let $f: [0, 1] \to \mathbb R$ be an $L^1$ function. Define for each $r > 0$, the blow up $f_r:[0, 1] \to \mathbb R$ by $$f_r (x) := \frac{f(rx)}{r}.$$ Suppose $f_r$ converges in $L^1$ to some ...
4 votes
3 answers
610 views

Mean width and perimeter

Does anyone know a simple, elementary and self-contained proof of the fact that the mean width of a convex two-dimensional body equals its perimeter divided by $\pi$?
2 votes
1 answer
179 views

Macroscopic sets - a notion of largeness for Lebesgue null sets

Let $E$ be a measurable subset of $\mathbb R$. We say $E$ is $\alpha$-macroscopic, for $0 \leq \alpha \leq 1$, if there exists an $\alpha$-Holder continuous function $f: \mathbb R \to \mathbb R$ such ...
1 vote
0 answers
81 views

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 ...
6 votes
0 answers
562 views

Is this result on the set of differentiability of the distance function to the fat Cantor set of any interest?

Quick summary: Consider the fat Cantor set $C$ of parameter $r$ for arbitrary $0 < r < \frac{1}{3}$, and the distance function to $C$, i.e. $D: [0, 1] \to \mathbb R$ given by $D(x) =\text{dist}(...
0 votes
0 answers
85 views

When can an affine functional on the dual be represented as an element of a Banach space?

In Measures Which Agree on Balls by Hoffmann-Jørgenson, we are given a functional $\varphi: T(x_0)\to (-\infty, \infty]$, which is a lower semicontinuous, affine, Baire function on a subspace $T(x_0)$ ...
1 vote
1 answer
158 views

Definition and properties of tangent functional

I am reading Measures Which Agree on Balls by Hoffmann-Jørgensen and I am somewhat confused. Here, $E$ is a Banach space, $S$ is the unit sphere, and $x \in S$. We let $\tau(x, \cdot)$ denote the ...
1 vote
2 answers
111 views

Computation of tangent functional

In Measures Which Agree on Balls by Hoffmann-Jørgenson, the tangent functional is defined as follows. If $x \in S$, we define the tangent functional $\tau(x,\cdot)$ at $x$ as \begin{equation} \...
1 vote
1 answer
182 views

Gateaux differentiability of the norm in Banach spaces

I'm struggling to understand a particular implication in the proof of Corollary 5 of this paper involving Gateaux differentiability of the norm. The claim is that Gateaux differentiability of the norm ...
1 vote
1 answer
82 views

Potentially elementary question on affine functions on Banach spaces

In Measures Which Agree On Balls by Hoffmann-Jørgensen, it is claimed that the function defined on $T(x)$, the set of normals to the unit sphere at $x$, given by $ \varphi(x^*) = \left\{ \begin{array}{...
3 votes
1 answer
128 views

Is every area-minimizing cone a level set of a least-gradient function?

Let $\mathbf{C}^n \subset \mathbf{R}^{n+1}$ be a minimizing cone with an isolated singularity. One example, in a space of even dimension, i.e. if $\mathbf{R}^{n+1} = \mathbf{R}^{2m}$, is the Simons ...
22 votes
1 answer
3k views

A gerrymandering problem - can you always turn a tie into a landslide victory?

Note: Here we use $|A|$ to denote the Lebesgue measure of a measurable subset $A$ of $\mathbb R^2$. Your party is running for election! In your country, voters are approximately uniformly distributed. ...
-1 votes
1 answer
122 views

What is an "open Baire set"?

In Measures Which Agree on Balls by Hoffmann-Jørgensen, it is stated that if $\varphi$ is a Baire function (which I presume means a pointwise limit of continuous functions), then $\{a<\varphi\}$ is ...
3 votes
3 answers
267 views

Extending a $C^1$ function on $\mathbb R^n$ to a set of finite $\mathcal H^{n-2}$ measure

Note: Here $\mathcal H^k$ denotes the $k$-dimensional Hausdorff measure. Let $n \geq 2$ be an integer, and $E \subset \mathbb R^n$ be a set of finite $\mathcal H^{n-2}$ measure. Suppose $f: \mathbb R^...
8 votes
3 answers
1k views

Almgren's mimeographed lectures notes on varifolds

I am trying to get some insights for the combinatorial argument of Pitts (in his PhD thesis 'Existence and regularity of minimal surfaces in Riemannian manifolds', Princeton University Press, 1981) to ...
3 votes
2 answers
170 views

Flat norm of currents and minimal surfaces

Let $A$ be a $k \leq n$ integral current with compact support over $\mathbb{R}^n$ (for conciseness). Its flat norm $F(A)$ can be defined via $ F(A) = \inf \{ M(T) + M(S) \, | A = T + \partial S \}$ ...
3 votes
1 answer
301 views

In which ways did geometric flows and variational methods from Riemannian geometry enter the symplectic world?

I am interested to learn about the role of geometric analytic methods for solving problems in symplectic geometry, In particular, I would like to know what results heavily rely on this machinery (incl....
0 votes
0 answers
80 views

Weak geometric lemma and Ahlfors-David regular boundary of a domain

It has been conjectured by David and Mayboroda (in the paper 'Approximation of Green functions and domains with uniformly rectifiable boundaries of all dimensions', while proving theorem 6.1 in the ...
1 vote
2 answers
202 views

If $\mathcal{H}^{n-1}(K)=0$ then $\mathcal{H}^n(K\times \mathbb{R})=0$

I am reading a paper Simon and Wickramasekera - A Frequency Function and Singular Set Bounds for Branched Minimal Immersions where the authors seem to claim that if $K\subset\mathbb{R}^n$ is a compact ...
2 votes
2 answers
161 views

When is the mode of a stochastic process a better statistic than the mean?

This is a soft question. I've been interested in Onsager-Machlup theory recently. Essentially, the Onsager-Machlup function serves the role of a density but it can exist on non locally compact spaces. ...
10 votes
1 answer
445 views

Isoperimetric inequality for closed curves in $\mathbb{R}^n$

A well known isoperimetric inequality for closed curves in $\mathbb{R}^2$ can be generalized to closed curves in $\mathbb{R}^{2n}$, see: https://mathoverflow.net/a/321505/121665. I have two questions: ...
2 votes
0 answers
162 views

A question from Leon Simon's "Lectures on Geometric Measure Theory"

In a book I am reading (Leon Simon, Lectures on Geometric Measure Theory) at some point the author claims that if a certain property $(P)$ holds for almost every $n$-plane $\pi\subset \mathbb{R}^{n+k}$...
1 vote
0 answers
53 views

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 ...
2 votes
1 answer
132 views

Minimal graph with confusing (?) property

Let $n \geq 2$ and $C = \{ (x,y) \in \mathbf{R}^{2n} \mid \lvert x \rvert = \lvert y \rvert \} \subset \mathbf{R}^{2n}$ be the Simons cone. (Whether this is area-minimizing or not does not seem to ...
0 votes
0 answers
123 views

What does $g_x^{-1}$ mean where $g$ is a Riemannian metric?

I am reading this paper about the Wasserstein-Fisher-Rao distance and they define the Wasserstein-Fisher-Rao distance on a manifold as follows (page 9): Here, $M$ is a compact Riemannian manifold, $\...
37 votes
0 answers
1k views

Converse of the Archimedean property of the sphere

In his remarkable book On the Sphere and Cylinder, where he came tantalizingly close to discovering calculus, Archimedes showed that the area of the portion of the sphere contained between a pair of ...
8 votes
1 answer
184 views

How do sets with unit fractional Hausdorff measure of dimension $>1$ look like?

Triggered by the recent question How can we not know the measure of the Sierpiński triangle? I would like to ask: Let $s>1$ and $s$ not be an integer. How to construct a set $A$ with $\mathfrak{H}^...
1 vote
1 answer
213 views

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 ...

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