Questions tagged [geometric-measure-theory]

The study of the geometric structure of measures, as well as the study of geometry from a measure-theoretic viewpoint, geometric measure theory has applications in partial differential equations, harmonic analysis, differential geometry, Riemannian geomerty, sub-Riemannian geometry, as well as calculus of variations. Statements such as the isoperimetric inequality and the coarea formula, and subjects such as the Plateau problem belong under this tag.

Geometric measure theory uses measure properties to study the geometric properties of sets in various spaces (usually Euclidean space).

Here are some key notions:

1) The Coarea formula expresses the integral of a function over an open set in Euclidean space, and is an adaptation of Fubini's theorem to geometric measure theory

2) Radon Measures are a type of measure on the $\sigma -$algebra of Borel sets of a Hausdorff topological space that is finite locally and inner regular. Radon measures are sets with the least 'regularity' required to approximate tangent spaces

3) There is the concept of the varifold, which is a measure-theoretic form of a differentiable manifold. It does this by maintaining the algebraic structure, but replacing any differentiability requirements with ones provided by rectifiable sets (these are sets that are smooth in the measure-theoretic sense).

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To show that the set point distant by 1 of a compact set has Lebesgue measure $0$

Could any one tell me how to solve this one? Let $K$ be a compact subset of $\mathbb{R}^n$, and $$A:=\{x\in\mathbb{R}^n:d(x,K)=1\}.$$ Show that $A$ has Lebesgue measure $0$. Thank you!
Marso
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Which sets are removable for holomorphic functions?

Let $\Omega$ be a domain in $\mathbb C$, and let $\mathscr X$ be some class of functions from $\Omega$ to $\mathbb C$. A set $E\subset \Omega$ is called removable for holomorphic functions of class $\mathscr X$ if the following holds: every function…
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Prove or disprove a claim related to $L^p$ space

The following question is just a toy model: Let $f:[0,1] \rightarrow \mathbb{R}$ be Lebesgue integrable, and suppose that for any $0\le a
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Manifolds with volume forms on every submanifold

If we equip a manifold with an inner product (i.e. we have a Riemannian Manifold) then we get a canonical volume form on that manifold (please mentally insert the prefix "pseudo" into my question whenever you feel it necessary). Whenever we have a…
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Has the notion of having a complex amount of dimensions ever been described? And what about negative dimensionality?

The notion of having a number $a \in \mathbb{R}_{\geq 0} $ associated to any metric space is described by the definition of a "Hausdorff Dimension". I was wondering if work has been done on spaces which (seem to) have a complex amount dimensions…
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Measure theoretic definition of curl

Is there a good measure theoretic definition of curl? To give an idea of the sort of equation that I'm looking for, here's now I define grad and div. For the gradient, say we are given a Fréchet differentiable function $f:X\rightarrow\mathbb{R}$,…
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Uncountable sets of Hausdorff dimension zero

Let $A \subset \mathbb{R}$ be a countable set. It is easy to see that $A$ has Hausdorff dimension $\dim_H(A) = 0$. Do there exist uncountable sets $A \subset \mathbb{R}$ with $\dim_H(A) = 0$?
JavaMan
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Best textbook for Geometric Measure Theory

I was wondering what is the best textbook for Geometric Measure Theory for self study. I am looking for one that isnt excessively detailed or long either as I found Rana's Introduction to measure theory fairly slow paced and superfluous to my…
Trajan
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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 equations are always true" p.361 4.1.8, or "I do not…
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$f : \mathbb{R} \to \mathbb{R}$ (Lipschitz) continuous implies $f(A)$ is Borel for all Borel $A$.

Full question: Let $(\mathbb{R}, \mathfrak{M}, m)$ denote the measure space $\mathbb{R}$ equipped with the Borel $\sigma$-algebra and the Lebesgue measure. Suppose $f : \mathbb{R} \to \mathbb{R}$ is Lipschitz continuous with Lipschitz constant $L$.…
mlg4080
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Sub-dimensional linear subspaces of $\mathbb{R}^{n}$ have measure zero.

I would appreciate it if someone could refer me to a proof (or simply give one here) for the statement in the title. That is: If $k
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2-dimensional Lebesgue measure of certain sets in $R^3$

Let $\theta >0$ and $E \subseteq \mathbb{R}^3$ be a closed set (I've added closedness as a new requirement) which satisfies the following condition: For any $x \in E$, there exist at least two lines $L_1,L_2 \subseteq E$ passing through $x$ with…
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Minimum area contained between measurable set and translate by $\lambda$: A strengthening of 2018 USA TSTST #9

Question Given $\lambda\in\mathbb{R}^+$, what is the smallest possible $c$ for which, given any measurable region $\mathcal{P}$ in the plane with measure $1$, there always exists a vector $\mathbf{v}$ with magnitude $\lambda$ so that the area shared…
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Can a set of Hausdorff codimension 2 disconnect a connected open set?

Consider a connected open set $U\subset \Bbb R^n$ (or a Riemannian manifold if you're ambitious), and $S\subset U$ closed and with Hausdorff dimension $\le n-2$. Is $U\setminus S$ connected? If not, does $\dim S\le n-3$ work? What is the optimal…
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Is there an open subset $A$ of $[0,1]^2$ with measure $>\frac{1}{100}$ that satisfies this property?

Can we find for any given $\varepsilon>0$ an open subset $A\subseteq[0,1]^2$ with measure $>\frac{1}{100}$ such that, for any smooth curve $\gamma:[0,1]\to\mathbb{R}^2$ of length $1$, the set $\gamma+A=\{\gamma(t)+a;t\in[0,1],a\in A\}$ does not…
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