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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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Ham sandwich theorem for integrable functions?

The classical ham sandwich theorem says that given $n$ measurable sets in $\mathbb{R}^n$, it is possible to divide all of them in half (with respect to their measure, i.e. volume) with a single $(n − 1)$-dimensional hyperplane. Does the theorem…
Mark
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Is a Jordan curve encircling a finite-perimeter set rectifiable?

Let $\gamma:[0,1]\rightarrow \mathbb R^2$ be a (continuous) simple closed curve (Jordan curve). The curve is not assumed to be rectifiable, i.e. we don't assume a priori that the length of the…
guestDiego
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A covering argument for metric Jacobian

Given a Lipschitz map between Carnot Groups $ f : G_1 \to G_2$, with homogeneous dilations $ \delta^1_s, \delta^2_s$, we have the almost everywhere Pansu derivative $ D_H f(x)(y) = \lim_{s\to 0} \delta^2_{1/s}[f(x)^{-1}f(x\delta^1_s y)] $. With $Q$…
smiley06
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Excess of Caccioppoli sets

I have a question concerning the excess of Caccioppoli sets: Given a Caccioppoli set $E\subset\mathbb{R}^{n}$, and let $A\in\mathbb{R}^{n}$ be open and bounded, we define according to Giusti (see http://www.springer.com/us/book/97808176315360)…
Alex
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Convergence of a sequence of characteristic function

I am reading something on BV functions and I am totally stucked with some assertion made during the development of one proof. How can I prove the following? Let $\varOmega \subset \mathbb{R}^n$ be an open set and $\{E_j\}_{j \in \mathbb{N}} \subset…
J. Salieri
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Measure on Locally Compact, Separable metric space

Simon (Geometric Measure Theory) says: If $X$ is a locally compact, separable metric space and $\mu(K) < \infty$ for all $K$ compact, then $X=\cup_{i=1}^\infty U_i$ where $U_i$ are open and $\mu(U_i) < \infty$. I tried showing this by using…
jpv
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Approximation of rectifiable sets

Let us say that I have a $H^d$ measurable, $H^d$ finite set $E \subset \mathbf{R}^n$ such that there exists a sequence of $C^1$ submanifold $(S_i)$ of $\mathbf{R}^n$ such that \begin{equation} H^d(E \setminus \bigcup_i S_i) = 0. \end{equation} Then…
captunus
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A question about Hausdorff measure

Let $ \mathscr{H}^m $ be the m-dimensional Hausdorff measure in $ R^n $. Let $ m
user55449
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The closure of an open subset in $\mathbb{R}^d$ is Ahlfors regular?

I have a question about Ahlfors regular space. Let $U$ be a bounded open subset in $\mathbb{R}^d$. We denote by $m$ the Lebesgue measure on $U$. Then, can we show the following? There exists a positive constant $C>0$ such that $$C^{-1}r^d \le m(U …
sharpe
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Estimate on the Hausdorff dimension of boundary of balls

I am reading Evans and Gariepy's book on GMT and I have a couple questions: 1) if E is a set of locally finite perimeter, is it true that E is $ \| \partial E\|$- measurable? 2) At a certain point, he uses the estimate $ \mathcal{H}^{n-1} ( \partial…
Jessica
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Polar coordinates, bounded domain with $C^{1}$ boundary

I have a question about a integral on a surface. It is well known that for any Integrable function $f$ defined on $\mathbb{R}^{n}$, it holds that \begin{equation} (1) \quad \frac{d}{dr} \int_{B(0,r)}f\,dm=\int_{\partial B(0,r)}f\,d \sigma \quad…
sharpe
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$ 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 $, $\partial ^{m}E$ is the measure theoretical boundary of E, and $\partial ^{*}E$ is the reduced boundary of E. From…
student
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How to get a compact subset where some conditions hold uniformly?

I'm reading Mattila's book "Geometry of sets and measures in Euclidean spaces". At the p. 222 in proof of Theorem 16.2 we have the following proposition: Let $\varepsilon >0$. Since E ($\mathcal H^m(E) < \infty$) has positive lower density…
Danil
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Why is $\mu(A \cap B_\rho(x)) \geq \limsup_{y \to x} \mu(A \cap B_\rho(y))$

Let $\mu$ be an outer measure on $X$. Let $A \subset X$. $x \in X$. Let $B_\rho(x)$ be the ball of radius $\rho$ centered at $x$. I read that if all Borel sets are $\mu$-measurable, and $\mu(B_\rho(x)) < \infty$, then $$ \mu(A \cap B_\rho(x)) \geq…
user35687
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Tangent measure to $\delta_y$.

I don't know anything about geometric measure theory but I was watching this video to try to understand the concept of tangent measures. I read that if $\mu=\sum c_i\delta_{y_i}$, then $Tan(\mu,a)=\{c\delta_0:c>0\}$. So I try to prove this. for…
roi_saumon
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