# 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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### A strange Lipschitz function

Let $n \geq 3$. Does there exist a Lipschitz function $f: \mathbb R^n \to \mathbb R$ such that the following conditions hold? The origin is a weak Lebesgue point of $\nabla f$, in the sense that the ...
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### Domains of type (A) are Lipschitz?

In this article and in the book of Ladyzhenskaya et al (1968) - Linear and Quasilinear Elliptic Equations we have the following definition of what is a domain of type (A): There is no example of a ...
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### Connectedness of Space of Caccioppoli Sets?

Let $(M^n, g)$ a closed, connected riemannian manifold. Is the space of all caccioppoli sets on $M$ connected with respect to the flat norm? How about with respect to the F norm (Flat metric on the ...
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1 vote
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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 $Y \subset \mathbb{R}^m$, ...
1 vote
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### When is a $1$-varifold $V$ the associated varifold of the reduced boundary of some Caccioppoli set?

Let $v_1$, $v_2$, $\cdots$, $v_l\in\mathbb{R}^n$ be unit vectors, $\mathbb{R}_v^+:=\{\lambda v:\lambda>0\}\subset\mathbb{R}^n$ be the ray in $v$'s direction; $n_1$, $n_2$, $\cdots$, $n_l>0$ be ...
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### Is there a singular function that is Hölder continuous of every order less than $1$?

We say a non-constant function $f$ on $[0, 1]$ is singular if it is continuous, and in addition differentiable almost everywhere with $f' = 0$ a.e. Does there exist a singular function that is Hölder ...
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### Is the $W^{1, \infty}$ limit of differentiable functions also differentiable?

Let $f_n$ be a sequence of differentiable functions on $[0, 1]$ with $f_n \to f$ uniformly for some (necessarily) continuous $f$. $f'_n - g \to 0$ in $L^{\infty}$ for some measurable $g$. Is it true ...
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### Is the $W^{1, \infty}$ limit of differentiable a.e. functions also differentiable a.e.?

Let $f_n$ be a sequence of continuous, differentiable a.e. functions on $[0, 1]$ with $f_n \to f$ uniformly for some continuous $f$. $f'_n - g \to 0$ in $L^\infty$ for some measurable $g$, where we ...
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### 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 ...
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### 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 ...
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### $\mathscr{H}^{n-2}(\Sigma)< \infty$ 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 ...
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### 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 ...
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1 vote
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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 ...
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### 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 $T(x_0)$ ...
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### 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 \...
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