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.
827 questions
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Volumes of n-balls: what is so special about n=5?
I am reposting this question from math.stackexchange where it has not yet generated an answer I had been looking for.
The volume of an $n$-dimensional ball of radius $R$ is given by the classical ...
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A kaleidoscopic coloring of the plane
Problem. Is there a partition $\mathbb R^2=A\sqcup B$ of the Euclidean plane into two Lebesgue measurable sets such that for any disk $D$ of the unit radius we get $\lambda(A\cap D)=\lambda(B\cap D)=\...
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Applications of Rademacher's Theorem
Rademacher's Theorem (that every Lipschitz function on $\mathbb{R}^{n}$ is almost everywhere differentiable) is a remarkable result on the structure of the space of Lipschitz functions, but I was ...
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Converse of the Archimedean property of the sphere
In his remarkable book On the Sphere and Cylinder, where he came tantalizingly close to discovering calculus, Archimedes showed that the area of the portion of the sphere contained between a pair of ...
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What are the obstructions for a Henstock-Kurzweil integral in more than one dimension?
I have recently come across the book The Kurzweil-Henstock Integral and its Differentials by Solomon Leader, in which, if I understand correctly, the HK integration process is modified in a way that ...
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Is arbitrary union of closed balls in $\mathbb{R}^n$ Lebesgue measurable?
Is an arbitrary union of non-trivial closed balls in the Euclidean space $\mathbb{R}^n$ Lebesgue measurable? If so, is it a Borel set?
@George
I still have two questions concerning your sketch of ...
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Rademacher theorem
If $f:\mathbb{R}^n\to\mathbb{R}^m$ is of class $C^1$ and $\operatorname{rank} Df(x_o)=k$, then clearly $\operatorname{rank} Df\geq k$ in a neighborhood of $x_o$. It is not particularly difficult to ...
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How can we not know the $s$-measure of the Sierpiński triangle?
I'm preparing a presentation that would enable high-school level students to grasp that the (self-similarity) dimension of an object needs not be an integer. The first example we look at is the ...
- 608
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The dual of $\mathrm{BV}$
$\DeclareMathOperator\BV{BV}\DeclareMathOperator\SBV{SBV}$I'm going to let $\BV := \BV(\mathbb{R}^d)$ denote the space of functions of bounded variation on $\mathbb{R}^d$. My question concerns the ...
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A gerrymandering problem - can you always turn a tie into a landslide victory?
Note: Here we use $|A|$ to denote the Lebesgue measure of a measurable subset $A$ of $\mathbb R^2$.
Your party is running for election! In your country, voters are approximately uniformly distributed. ...
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Existence of subset with given Hausdorff dimension
Let $A\subseteq \mathbb{R}$ be Lebesgue-measurable and let $0<\alpha<1$ be its Hausdorff dimension.
For a given $0<\beta <\alpha$ can we find a subset $B\subset A$ with Hausdorff ...
- 1,127
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Are functions of bounded variation a.e. differentiable?
In general, it is well known that, on the real line, say on $[0,1]$, if a function $f$ is of (pointwise) bounded variation, meaning that
$$
\sum_{i=1}^n |f(x_i)-f(x_{i-1})| <+\infty
$$
for every ...
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Best Hölder exponents of surjective maps from the unit square to the unit cube
The Peano's square-filling curve $p:I\to I^2$ turn's out to be Hölder continuous with exponent $1/2$ on the unit interval $I$ (a quick way to see it, is to note that $p$ is a fixed point of a ...
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22
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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 ...
- 1,636
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Image of the trace operator
It is well-known that we have the trace theorem for Sobolev spaces. Let $\Omega$ be an open domain with smooth boundary, we know that the map
$$ T: C^1(\bar\Omega) \to C^1(\partial\Omega) \subset L^...
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