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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 ...
Piotr Hajlasz's user avatar
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 ...
Mizar's user avatar
  • 3,126
21 votes
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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 $\sum_{i = 1}^{(1-\epsilon)n} 2^i < 2^{(1-\epsilon)n+1}$ ...
Ronnie Pavlov's user avatar
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 ...
Jacob Manaker's user avatar
20 votes
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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 ...
Saúl RM's user avatar
  • 10.4k
19 votes
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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 ...
Pietro Majer's user avatar
  • 58.1k
17 votes
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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 $\infty$ ...
Nate River's user avatar
  • 5,745
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 ...
Camillo De Lellis's user avatar
14 votes
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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,\...
Piotr Hajlasz's user avatar
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 ...
ThiKu's user avatar
  • 10.3k
14 votes
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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 ...
Piotr Hajlasz's user avatar
14 votes
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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 ...
Skeeve's user avatar
  • 1,277
14 votes

Generalized Stokes' theorem

Harrison, Jenny, Stokes' theorem for nonsmooth chains. Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 2, 235–242. This research announcement reports progress in developing a viable theory of integration ...
Ben McKay's user avatar
  • 26k
12 votes
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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 $\{\...
Piotr Hajlasz's user avatar
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 ...
Piotr Hajlasz's user avatar
12 votes
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Unknown work of Nöbeling on topological/Hausdorff dimension

So, the sought for paper is: Nöbeling, G., Hausdorffsche und mengentheoretische Dimension, Ergebnisse math. Kolloquium Wien 3, 24-25 (1931). And here is a ``translation" (to English and to modern ...
Behnam Esmayli's user avatar
12 votes
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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 ...
Nik Weaver's user avatar
  • 42.4k
12 votes
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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 ...
Sam Forster's user avatar
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{...
Nate River's user avatar
  • 5,745
11 votes
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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 $\{0\}$ ...
Igor Belegradek's user avatar
11 votes
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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 ...
user135139's user avatar
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 $y,z$ ...
fedja's user avatar
  • 60.9k
11 votes
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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 ...
Kostya_I's user avatar
  • 8,947
11 votes
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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 ...
Wojowu's user avatar
  • 27.7k
11 votes
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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 ...
Saúl RM's user avatar
  • 10.4k
11 votes
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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 $j\in[-1,1]$ ...
Saúl RM's user avatar
  • 10.4k
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 $n$, ...
Gerald Edgar's user avatar
  • 40.6k
10 votes
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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 "...
coudy's user avatar
  • 18.6k
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_\...
Martin Kell's user avatar
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 ...
Gabe K's user avatar
  • 5,494

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