45
votes

### Applications of Rademacher's Theorem

I will mention seven different applications:
Characterization of almost everywhere differentiability.
The following result is a consequence of the Rademacher theorem:
Theorem (Stepanov). A function ...

24
votes

Accepted

### Are functions of bounded variation a.e. differentiable?

No. Take a dense countable set $\{x_1,x_2,\dots\}$ in $\mathbb{R}^d$ and a sequence $(r_i)\subseteq\mathbb{R}^+$ such that $\sum_i r_i^{d-1}<\infty$. Then the function
$$f=1_{\bigcup_{i=1}^\infty ...

21
votes

Accepted

### Why do almost all points in the unit interval have Kolmogorov complexity 1?

I'm not an expert on Kolmogorov complexity, but this does seem like a counting argument: for any fixed $\epsilon > 0$, there are only $\sum_{i = 1}^{(1-\epsilon)n} 2^i < 2^{(1-\epsilon)n+1}$ ...

20
votes

### Generalized Stokes' theorem

When I wrote that Wikipedia paragraph, I think I had in mind Theorem 5.16 in Evans & Gariepy, Measure Theory and the Fine Properties of Functions, which essentially proves Stokes' theorem for ...

20
votes

Accepted

### A gerrymandering problem - can you always turn a tie into a landslide victory?

Yes, the almost partition exists. Instead of letting $\mu(E)\geq\frac{\mu(\Omega)}{2}$, I let $\mu(E)\in(0,\mu(\Omega))$ be arbitrary and proved that you can divide $\Omega$ into $N$ open simply ...

19
votes

Accepted

### Is the $W^{1, \infty}$ limit of differentiable functions also differentiable?

This is true, $f$ is actually everywhere differentiable. It is the "Limit under the Sign of Derivative" Theorem; it also holds for sequences of maps between Banach spaces (and you may even ...

17
votes

Accepted

### How badly can the Lebesgue differentiation theorem fail?

Metafune has given an example of the limit failing to be $0$ at a particular point - namely for $n > 1$, the function $|x|^{-\alpha}$, with $1 \leq \alpha < n$ has that limit equal to $\infty$ ...

16
votes

### Almgren's mimeographed lectures notes on varifolds

I just bumped into your post after doing some google search to find a bibtex entry for Almgren's notes.
I have a copy of them: I could copy it and send it to you (or maybe scan it and share it via ...

14
votes

Accepted

### Can $C^1$ mappings with derivative of low rank be approximated by smooth maps?

There is a counterexample.
Example. There is $f\in C^1(\mathbb{R}^5,\mathbb{R}^5)$ with ${\rm rank}\, Df\leq 3$ that cannot be approximated in the supremum norm by mappings
$g\in C^2(\mathbb{R}^5,\...

14
votes

### Continuous deformation of soap films

You are asking whether a least area surface depends continuously on its boundary curve, or in mathematical terms: whether for every curve $\gamma$ there exists a least area surface $S$ with $\partial ...

14
votes

Accepted

### Is the intersection of two Caccioppoli (i.e. finite perimeter) sets Caccioppoli?

That is true. Caccioppoli sets are also known as sets of finite perimeter.
Theorem. Suppose $f\in L^1(\mathbb{R}^n)$ vanishes outside the unit cube $[0,1]^n$. For $i=1,2,\ldots,n$ consider the ...

14
votes

Accepted

### Existence of subset with given Hausdorff dimension

First of all, $\dim_{H} (A) = \alpha$ iff $ H^k(A)=\infty$ for all $k<\beta$ and $H^k(A) = 0$ for all $k>\beta$. Then $H^\alpha(A) = \infty$ for all $\alpha \in (0,\beta)$.
If $A$ is closed ...

14
votes

### Generalized Stokes' theorem

Harrison, Jenny, Stokes' theorem for nonsmooth chains.
Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 2, 235â€“242.
This research announcement reports progress in developing a viable theory of integration ...

12
votes

Accepted

### Is there a non-atomic finite positive measure in the plane, of which uncountably many projections have atoms?

Such a measure cannot exist. Suppose to the contrary that we have an uncountable family of lines $\ell$ such that $\mu(\ell)>0$. Then there is $\epsilon>0$ and an infinite family of lines $\{\...

12
votes

### Existence of subset with given Hausdorff dimension

The answer is yes under the additional assumption that the set is compact and I do not know what happens in the general case. The result is a consequence of the following one, see [1] and references ...

12
votes

Accepted

### Unknown work of Nöbeling on topological/Hausdorff dimension

So, the sought for paper is:
NĂ¶beling, G., Hausdorffsche und mengentheoretische Dimension, Ergebnisse math. Kolloquium Wien 3, 24-25 (1931).
And here is a ``translation" (to English and to modern ...

12
votes

Accepted

### On functions with strict Lipschitz constant

I guess it suffices to give an example for $n = 1$. If $f: \mathbb{R} \to \mathbb{R}$ is an example then $g(x_1, \ldots, x_n) = f(x_1)$ will be an example for any $n \geq 1$.
All we need is a ...

12
votes

Accepted

### If a function $f$ is $(1+\varepsilon)$-times Lebesgue differentiable everywhere, is $f$ a constant function?

I realise I'm bumping into you again and already gave you an answer elsewhere after you posted this, but I thought I'd post my answer here for others to see. The answer is yes, $f$ has to be constant ...

12
votes

Accepted

### Does every differentiable a.e. function admit a maximally differentiable representative?

$\DeclareMathOperator*\appliminf{app-liminf}\DeclareMathOperator*\applimsup{app-limsup}\DeclareMathOperator*\applim{app-lim}\DeclareMathOperator*\essliminf{ess liminf}\DeclareMathOperator*\esslimsup{...

11
votes

Accepted

### Proper homotopy

The following counterexample is stolen from page 1 of this paper by Thomas Rot. The map $[0,1]\times\mathbb R\to\mathbb R$ given by $(t,x)\to (1-t)x^2+x$ is not proper, e.g., the preimage of $\{0\}$ ...

11
votes

Accepted

### Metric measure spaces: in what sense is analysis on these spaces "non-smooth"

I highly recommend the survey article "Nonsmooth Calculus" by Juha Heinonen, available here.
The beginning of the introduction reads:
"The word nonsmooth in the title refers both to functions and ...

11
votes

Accepted

### A Besicovitch-type Covering Theorem

This trivial counterexample in $\mathbb R^2$ should have taken me five minutes. Instead, I spent almost two days. The moral is the usual one: after 50 you'd better give up on mathematics.
Let $y,z$ ...

11
votes

Accepted

### Is the composition of two nowhere differentiable functions still nowhere differentiable?

The composition may have points of differentiability.
Let $f_0(x)=x$ for $x\geq 0$ and $f_0(x)=2x$ for $x<0$. Let $g_0(x)=2x$ for $x\geq 0$ and $g_0(x)=x$ for $x<0$. Then none of them is ...

11
votes

Accepted

### Tiling the plane with finitely many congruent pieces

Write $A_i=T_i(A)$ for $A=A_1$, where each $T_i$ is a rigid motion. For each $i$, we have $|A_i\cap B(r)|=|T_i(A\cap T_i^{-1}(B(r))|=|A\cap T_i^{-1}(B(r))|$. The symmetric difference between this set ...

11
votes

Accepted

### If $\mathcal{H}^{n-1}(E)=0$ then $\mathbb{R}^n\setminus E$ is connected

Yes, $\mathbb{R}^n\setminus E$ has to be path-connected.
Let $x,y\in\mathbb{R}^n\setminus E$, we will prove that there are many paths going from $x$ to $y$ inside $\mathbb{R}^n\setminus E$. We can ...

11
votes

Accepted

### Uncountable collections of distinct subsets of an interval (existence)

My comment reposted as an answer:
If the continuum hypothesis holds, then we can give a well order $\prec$ to $\mathbb{R}$ isomorphic to the first uncountable ordinal. And then for each $j\in[-1,1]$ ...

10
votes

### Fractals of dimension zero

One example: the set of
Liouville numbers has Hausdorff dimension zero.
In number theory, a Liouville number is an irrational number $x$ with the property that, for every positive integer $n$, ...

10
votes

Accepted

### Ergodicity and mixing of geodesic and horocyclic flows

The algebraic approach is just one of many ways to deal with the geodesic and horocycle flows. Also your quick summary does not really pay tribute to the many ways to deal with it using "...

10
votes

### Derivative of distance function to a closed, rectifiable set

The distance $f: x \mapsto \mathsf{dist}(x,\Gamma)$ to a closed set $\Gamma$ in $\mathbb{R}^n$ is differentiable in $x \notin \Gamma$ iff the nearest point projection is unique; denote this by $x_\...

10
votes

Accepted

### A better version of Weyl's Law or uniform estimates of Laplacian higher eigenvalues

It seems unlikely that such a bound holds except in very special cases. For instance, it fails for round spheres, which have very large multiplicity of eigenvalues. In fact, for spheres the eigenvalue ...

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