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 and Riemannian geometry, as well as calculus of variations. Statements ...
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230 views
Estimate the surface area of a 2D shape where the only known value is the length of the enclosing boundary
Wondering if it is possible to estimate the surface area of a 2D shape where the only known value is the length of the enclosing boundary, and that it is know the internal surface area is solid. ...
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0answers
149 views
Corollary of Lebesgue decomposition theorem and counter-example
Refferring to the Lebesgue decomposition theorem in Lebesgue decomposition theorem and fundamental theorem of calculus there is a corollary when the measure is the Lebesgue measure that states: if ...
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1answer
106 views
Surface measure for Lipschitz domain
Let $D\subset R^d$ be a bounded Lipschitz domain.
Must there exist a bounded function $\Phi$ on $\partial D$ and collections of subsets $(\partial D )^{\epsilon} \subset \partial D $ (indexed by ...
3
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0answers
76 views
Radon-Nikodym derivative of the Hausdorff measure transform
Let $H^{m}$ be the $m$-dimensional Hausdorff measure, let $D = \operatorname{diag }(d_1,...,d_k)$ be a nonsingular scale matrix. Consider the change of measure formula:
$$
\int\limits_{A} f(Dx) \; ...
5
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1answer
190 views
Is the Hausdorff outer measure regular?
An outer measure $\mu^*$ is said to be regular if for every set $A \subset X$
$$\mu^\ast (A)=\inf\{\mu^*(E) : E\supset A \text{ is } \mu^\ast\text{-measurable} \}$$
To check that an outer ...
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1answer
84 views
The modulus of curve family
I'm reading book "Geometric function theory and non-linear analysis" and there is one thing that I did not understand.
A curve family $\Gamma$ is a collection of curves. An admissible
density is ...
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1answer
86 views
Dimension invariance
Where can I find the Theorem of invariance of the dimension with diffeomorphisms? And about Between Homeomorphism?
I also want to know about the Hausdorff dimension invariance what is deeper!
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0answers
99 views
Co-area formula involving non-integer Hausdorff measure
Is there any co-area formula involving non-integer Hausdorff dimension?
Moreover is it sensible to write the following:
Let $S$ be a subset in $ \mathbb{R}^n$ with Hausdorff dimension s ...
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59 views
Isoperimetric inequalities with relative perimeter
It is a well known result that if $\Omega\subset \Bbb{R}^N$ is an open set, with regular boundary (smooth, or Lipschitz) then the problem
$$ \min_{E \subset \Omega |E|=c} Per(E)$$ has a solution, ...
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1answer
84 views
Reference for Radon measure
Can anyone provide references for Radon measures, Bounded variation functions space, and Lebesgue differentiation theorem for Radon Measures, Hausdorff dimensions?
Note:
Online references will be ...
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3answers
271 views
Total variation of (weakly) differentiable functions
the total variation of a function $u\in L^1(\Omega)$, $\Omega\subset \mathbb{R}^n$, can be defined as
$$
\sup \{ \int_\Omega u \; \mathrm{div} g \; dx:\; g \in C_c^1(\Omega,\mathbb{R}^n), \; \lvert ...
5
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2answers
176 views
Change of variable within an integral of the Hausdorff measure
Let $T \colon \mathbb{R}^n \to \mathbb{R}^n$ be a linear map, $H^{m}$ be a Hausdorff measure.
Is it true that
$$
\int\limits_{T(M)} f(x) H^{m}(dx) = |\det{T}| \int\limits_{M} f(T(x)) H^{m}(dx)
$$
...
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1answer
125 views
Volume form and Hausdorff measure
Let $M$ be a smooth orientable $(n-1)$-dimensional submanifold in $\mathbb{R}^n$, $dS$ be its volume form and $dH^{n-1}(x)$ be an $(n-1)$-dimensional Hausdorff measure. How to show than that
$$
...
6
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2answers
336 views
Relations between various definitions of a Radon measure
The following various definitions of a Radon measure seem to be given for the Borel sigma algebra of different types of topological spaces: general, Hausdorff, locally compact, or locally compact ...
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4answers
231 views
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 ...
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1answer
244 views
Banach-Tarski Paradox on the middle third Cantor set
In analysis and topology, the middle third Cantor set $C$ is often a very interesting topic of discussion. My question is that is it possible to have some sort of measure preserving bijection $f: ...
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1answer
235 views
Lebesgue measure on normal matrices
Consider the space of $n\times n$ complex matrices, and equip it with its Lebesgue measure $dX$, seen as a $2n^2$-dimensional real vector space [edit: or better, a complex vector space (see the answer ...
8
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1answer
254 views
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$?
5
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2answers
148 views
Methods to define areas
I always thought that areas are defined by integrals, until I read Michael Spivak's Calculus p.289:
The desire to define area was the motivation, both in this book and historically, for the ...
2
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3answers
81 views
Hausdorff measure with non-power test function
At my analysis course some time ago we were told that there is definition of Hausdorff measure through the test functions which are continuous and non-decreasing $h:(0,\infty)\to(0,\infty)$ and ...
3
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1answer
271 views
How to understand currents in geometric measure theory?
I find it is hard to catch the current, sometimes it is just the picture as its support set (if I do not miss it).
What is the heart idea of the current? What are the benefits to introduce such an odd ...
6
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0answers
204 views
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 − ...
6
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2answers
208 views
Symmetry of Solution to Classical 3-Dimensional Isoperimetric Problem
A while ago I attempted to solve the classical isoperimetric problem in 3-dimensions, namely "Find the surface that has the smallest surface area for a given volume".
At that time for me to write ...
2
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3answers
314 views
How can a density be larger than $1$?
From Frank Morgan: Geometric Measure Theory, Fourth Edition: A Beginner's Guide, page 13,the $2$-dimensional density of the cone $x^2+y^2=z^2$ at $0$ is $\sqrt{2}$. I feel strange of that，roughly ...
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199 views
How do we prove that a sphere maximizes the volume enclosed among all simple closed surfaces of given surface area?
How do we prove that among all closed surfaces with a given surface area, the sphere is the one that encloses the largest volume, and not do it by cases?
so far I've tried is that I know the formula ...
3
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1answer
308 views
Calculating the upper Minkowski dimension of the set $\{0,1,\frac{1}{2}, \frac{1}{3}, \ldots \}$
The upper Minkowski dimension of a compact set $A$ in $\mathbb{R}$ is defined as
$$
\overline{\dim}_M = \inf \{ \epsilon > 0 : \text{ there is a constant } C(\epsilon) \text{ such that } ...
