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.
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A consequence of Huisken's rescaled monotonicity formula : Stone's Lemma
The context is that of the mean curvature flow, more precisely, concerning Type I singularities and the rescaling procedure. The text I am following is by Mantegazza: "Lecture notes on Mean ...
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Reference for currents in geometric measure theory - continuous version?
Let $M$ be a Riemannian manifold. I am interested in the dual of the space of continuous (not necessarily smooth) vector field on $M$.
The dual of the space of smooth vector field would be the space ...
- 1,381
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How to prove that rigid bodies have fixed areas during their movements?
a rigid body is a pair of: a set $S$ of points with at least two distinct points on space $\mathbb R^3$ and a special continuous position function for each point of $S$ $f:S \times [t_1,t_2] \to \...
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Compute the total variation of a linear bounded functional
Let $\mu$ be a Radon measure in $\mathbb R^n$, if $L:C_C^{0}(\mathbb R^n , \mathbb R^m) \to \mathbb R$ be a linear functional, the total variation of $L$ is defined by
$$|L|(A)= \sup \{ L(\varphi): \...
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A key lemma in the proof Besicovitch Covering lemma.
I want to prove this lemma which is left as exercise for the reader in the book Measure Theory and Integration of Micheal E.Taylor. The lemma is used to prove the Besicovitch Lemma. This is the ...
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An elementary decomposition of a countably $\mathcal{H}^k$-rectifiable set
Let $1 \le k \le n-1$. We say $M \subset \mathbb{R}^n$ is countably $\mathcal{H}^k$-rectifiable if $M$ is $\mathcal{H}^k$-measurable and there exist countably many Lipschitz maps $f_h : \mathbb{R}^k \...
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23
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About Frostman Lemma [closed]
Let $H^g$ denote the Hausdorff measure with respect to a gauge function $g$. Suppose $K \subset \mathbb{R}^d$ is compact. Is it true that $H^g(K)>0$ if and only if there exists a (regular Borel) ...
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Proof request: Extending vector valued function of unit outer normals to the whole space $R^n$
I was wondering if anyone could provide a proof or a reference of the following proposition:
Let $C$ be an open subset of $\mathbb{R}^n$ with $C^2$ boundary so that the unit outer normal vector $v(x)$ ...
- 387
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Integral of a function over the unit sphere [closed]
Let $S^N = \{ y = (\xi, \eta) \in \mathbb{R}^N : |y| = 1\}$ the unit sphere on $\mathbb{R}^N$. Define $f : S^N\backslash A \to \mathbb{R}$ by $$f(y) = \frac{1}{|\xi|^{a}|\eta|^b},$$
where $a,b \in \...
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What is isoepiphany in math? [closed]
Archimedes studied isoepiphany, later handled by an arabic scholar al Haytham
but I can't find a defintion?
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When is the "flat limit" of a submanifold not a submanifold?
We have a compactness result, where the "flat limit" of an integral current is itself an integral current. (with some conditions)
Now I am curious about submanifolds. I am expecting that an ...
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Determining the measure of $n$-dimensional space in $(n+1)$-dimensional space
I'd like to understand the geometric abstraction of measure in more depth, and my contemplations about it led me to a number of conclusions:
A geometric point is infinitely small in relation to any ...
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The Weak Besicovitch Covering Property and the Lebesgue Differentation Property
Some preliminary terminology.
Before making the question let me introduce some terminology.
Notation. Let $X$ be a set and $A$ a subset of $X$. I denote by $\chi_A\colon X\to\{0,1\}$ the ...
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Question about proof of Krantz proposition 1.6.14 (Steiner symmetrization preserves compacteness)
I was reading the introductary chapter on the book Geometric integration theory (Birkhäuser, Cornerstones series 2008, Krantz & Parks) and stumbled upon the following theorem/proposition about ...
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Steiner symmetrization preserves compactness？
I have read " Convexity, H.G.Eggleston, 1958'' and in page 91, theorem 43, it proves that a closed, bounded and convex set $\mathcal{X}$ is still closed bounded (and convex) under Steiner ...
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Real-Valued Error Function on SO(3)
In some geometric control papers, the author usually defines the real-valued error function to be:
$\Psi(R,R_d)$ = $\frac{1}{2} Trace[I - R_d^T R ]$. (1)
where $R_d$ is the arbitrary smooth attitude ...
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Looking for regularity classes which are weaker than continuity
To contextualize my question, recall that Lipschitz (i.e. $C^{0,1}$) functions by Rademacher's theorem are differentiable almost everywhere. It's interesting to note that this is sharp (and apparently ...
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How to calculate the area of the intersection of infinite union and finite set?
A line segment in $\mathbb{R}^2$ is defined as $\mathbb{L} = Conv\{\textbf{p}_1,\textbf{p}_2\}$. Obviously, its area is 0. Given a time-parameterized continuous trajectory $\textbf{T}(t)$ in $\mathbb{...
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Is there an Approximate tangent space that is NOT a Classical tangent space?
First given $M \subset \mathbb{R}^n$, and $x \in \mathbb{R}^n$, $r>0$, we define the blow up by:
$$ \Phi_{x,r}(y) = \frac{y-x}{r}.$$
We say $x \in \mathbb{R}^n$ has a $k$ dimensional approximate ...
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How to calculate the distance from the inside of a swept body to it's boundary?
As an example, a line segment on 2D plane, moving along a certain trajectory, produces a swept body. Select a point inside the swept body, how to get the distance from this point to the boundary of ...
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Space of smooth differential forms with compact support on $U \subset \mathbb{R}^n$ is separable
I am interested in how to prove that the space of smooth differential forms with compact support on some open $U \subset \mathbb{R}^n$, denoted $\mathcal{D}^n(U)$, is separable, i.e. it has a ...
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Federer's proof of the Area Formula
I went through the proof of Theorem 3.2.3(1) (the Area Formula) from Federer's GMT book. The statement of the theorem is tha following:
Suppose $f:\mathbb{R}^m \to \mathbb{R}^n$ is Lipschitzian with $...
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A lemma used to prove Besicovitch Theorem
Let $X$ be a metric space and $\mu\colon\mathcal{P}(X)\to[0,+\infty]$ an outer measure over $X$ such that all open subsets of $X$ are $\mu$-measurable (then $\mu$ is a Borel measure) and $\mu$ is ...
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A question about Theorem 2.8.2 of "Geometric Measure Theory" of Federer.
Let $X$ be a metric space and $\mu\colon\mathcal{P}(X)\to[0,+\infty]$ an outer measure over $X$ such that all open subsets of $X$ are $\mu$-measurable (then $\mu$ is a Borel measure) and $\mu$ is ...
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A question about Theorem 2.8.7 from "Geometric Measure Theory" of Federer
Let $X$ be a metric space and $\mu\colon\mathcal{P}(X)\to[0,+\infty]$ an outer measure over $X$ such that all open subsets of $X$ are $\mu$-measurable (then $\mu$ is a Borel measure) and $\mu$ is ...
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Reference request: book on abstract algebra
I'm having a hard time understanding most of the material covered in chapter 1 of Federer's GMT book (it includes tensor products, graded algebra, exterior algebra); this is mostly because I'm not ...
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Weak convergence of measures on the Gromov boundary of a finitely generated free group
In the set-up of my previous post, let $\theta$ be a purely non-atomic finite regular measure defined on the Borel $\sigma$-algebra of the metric space $(\partial F, d)$. We say $\theta$ admits a $\...
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What are some ways of measuring the divergence between n-lines
I was attempting to discern general patterns in electric data. I subsequenced the data at different tiers. One of which being intraweekly variation. I calculated the euclidean barycenters for each ...
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Help with Theorem 2.9.7 from Federer's Geometric Measure Theory
Suppose $\phi$ and $\psi$ are Borel regular measures (outer measures) on a metric space $X$ such that $\phi(A),\psi(A)<\infty$ for every bounded subset $A\subseteq X$. One defines a Borel regular ...
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Measure of Cylinder Intersecting Sphere in $\mathbb{R}^n$
Please check my work.
Let $\mu$ be the uniform measure on $\mathbb{S}^n$, the unit sphere with radius $1$, and $$A=\{x\in\mathbb{S}^n:x_1^2+x_2^2\le \sin^2\alpha\}$$
where $\alpha$ is constant.
Claim $...
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Writing an integral over a sphere in terms of integrals of another spheres with lower dimension
Suppose $N = 4$. Given $g \in L^{1}(S^{N-1})$, I would like to know if is it possible to write
$$
\int_{S^{N-1}} g(x,y,z,w) d\sigma^{N-1} = C \int_{S^{N-3}}\left( \int_{S^{N-3}} g(x,y, z,w) d \sigma^{...
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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 ($\...
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Relation between Hausdorff measure and Lebesgue Measure
I have the following definition of $\alpha$-dimensional Hasudorff measure
Let $E$ be subset of $\mathbb{R^n}$. Fix $\alpha \in [0, \infty)$, $\kappa \in \mathbb{R}^+$ and $\epsilon > 0$. Then $\...
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Lebesgue measure of the euclidean ball [closed]
The following is written in "An introduction to Geometric Measure theory" by F. Maggi.
Let $B$ be an open euclidean ball in $\mathbb{R}^n$ and $\omega_n=|B|$, where $|B|$ is Lebesgue measure ...
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Approximate tangent space agrees with tangent space of submanifold of $\mathbb{R}^n$
I am stuck on trying to prove that the approximate tangent space of a submanifold of $\mathbb{R}^n$ agrees with its tangent space. To make things more precise I'll give the relevant definitions. Note ...
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Hausdorff dimension of an algebraic variety
I have the following elementary question that I cannot quite figure out myself or find an appropriate reference:
Let $p$ be a nonzero homogeneous polynomial which we view as a function on $\mathbb R^n$...
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Application of Banach-Alaoglu theorem to extract convergent subsequence of currents
While reading about currents I came across the following lemma in Lectures on Geometric Measure Theory by Leon Simon on page 135:
Lemma. If $\left\{T_j\right\}_{j\in\mathbb{N}}$ is a sequence of ...
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Computing "by hand" the approximate tangent space of a particular set
I am trying to do the following exercise. Consider $E= \bigcup_{j=1} ^{\infty} I_j$ with $I_j$ one dimensional segments in $\mathbb{R}^n$ with lengths $l_j$ such that:
$\sum_{j=1} ^{\infty} l_j < \...
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Proof that a Lipschitz function is a.e. approximately differentiable by using it is "almost" $C^1$
I am trying to do the following exercise: to prove that a Lipshitz function $f: \mathbb{R}^n \mapsto \mathbb{R}^m$ is $\mathcal{L}^n$-almost everywhere approximately differentiable using the following ...
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Proving Lebesgue dominated convergence theorem using Egorov theorem
I'm trying to prove Dominated Convergence Theorem (DCT) by using Egorov's theorem. And so far, I know how to do it when the functions are defined in finite measure set in real. Is it possible to also ...
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Hausdorff measure and uniform subdivisions
Sorry I never got to learn about Hausdorff measures so this may come as simple. Suppose
$$ A = \{x\in [0,1)^d: f(x)=0\} $$
where $f$ is analytic and nontrivial. Then $A$ is an analytic variety of ...
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Is the Lebesgue measure of the boundary of a union of $n$-balls zero?
I'm working in $\mathbb{R}^n$, but I believe that the question is relevant in any metric space.
Let $B_1^n$ be the closed ball of radius 1 centered at the origin, and let $\oplus$ denote Minkowski ...
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If there is a zero measure set with positive measure pre-image, is there a level set with positive measure?
Let $f:\mathbb{R}\to\mathbb{R}$ be smooth (I suppose $C^1$ should be enough, but I would be happy with an answer when $f\in C^\infty$).
Assume that there exists a Borel set $A\subset\mathbb{R}$ with ...
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Can a Jordan curve contain measure-theoretic interior points of the domain it bounds?
Let $I$ be an interval, and $\gamma:I\to \mathbb{R}^2$ a Jordan curve. By the Jordan--Schoenflies Theorem $\gamma(I)$ splits up $\mathbb{R}^2$ in two connected pieces, that is, $\mathbb{R}^2\setminus \...
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Calculating PDF of a function of random variables via dirac delta integral
I came across some papers 1, 2 and others which use the following formula
$
p(y) = \int_{\mathcal{X}} \, d^{n}x \, \delta(F(\vec{x}) - y) \, p(\vec{x})
$
where $p(y)$ is the PDF of the variable ...
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How do we get $ J_{f \circ \psi} (x) = J_\psi(x) J_f(y) $ for $f$ on a submanifold, and $\psi$ is local coordinates?
The following is taken from Leon Simon Geometric Measure Theory: Let $f: M \to \mathbb{R}^P$ for $P \geq n$. Where $M$ is an $n$ dimensional smooth submanifold of $\mathbb{R}^{n+l}$, and $f$ is ...
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203
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Measurability of a classical topological surface and its measure
Let $\Sigma \subset \mathbb{R}^3$ be a set with the following property: Given any $p\in \Sigma$, $\exists$ $W_p \subset_{\text{open}} \mathbb{R}^3$, $U_p \subset_{\text{open}} \mathbb{R}^2$ such that $...
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52
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Understanding the proof of the Hausdorff area formula with multiplicities
We have the following theorem:
Let $f: A \subset \mathbb{R}^n \to \mathbb{R}^m$ be Lipschitz, with Lipschitz constant Lip $f$, $1 \leq n \leq m$. Then we have
$$ \int_{\mathbb{R}^m} H^0(f^{-1}(y) \cap ...
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Applying Hausdorff Area formula with multiplicity
I am trying to understand the following formula.
Let $f: A \subset \mathbb{R}^n \to \mathbb{R}^m$ be Lipschitz, with Lipschitz constant Lip $f$, $1 \leq n \leq m$. Then we have
$$ \int_{\mathbb{R}^m} ...
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1
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37
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Is a function with bounded local averages necessarily bounded?
Given a measurable function $f : \mathbb{R}^n \to \mathbb{C}$, we can define the Hardy-Littlewood maximal operator $M$ by
$$Mf (x) = \sup_{0 < r < \infty} \mu(B(x, r))^{-1} \int_{B(x, r)} | f | \...
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