All Questions
8 questions
23
votes
3
answers
873
views
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 ...
- 60.6k
9
votes
1
answer
637
views
Is there a characterization of the Hausdorff measures?
It is known that there is a unique measure on the Borel $\sigma$-algebra of $\mathbb{R}^n$ such that the measure of the rectangle $\prod_i [a_i,b_i[$ is $\prod_i (b_i-a_i)$. This is the Lebesgue ...
- 1,035
8
votes
1
answer
344
views
How large can the set of turbulent points be?
This question resisted attempts on MSE.
Let $E \subset \mathbb R^n$ be a Lebesgue measurable set. We say that $x \in \mathbb R^n$ is a turbulent point of E if both the following conditions hold:
$$\...
- 6,313
7
votes
3
answers
548
views
Maximal Hausdorff dimension of the set on which derivatives do not agree
Let $f, g: [0, 1] \to \mathbb R$ be functions that are differentiable a.e. with $f’ = g’$ almost everywhere. What is the supremal Hausdorff dimension of the set on which $f$ and $g$ are both ...
- 6,313
6
votes
0
answers
822
views
Is the $L^\infty$ norm of the derivative the same under the Hausdorff and Lebesgue measure?
Note: Here $\mathcal H^k$ denotes the $k$-dimensional Hausdorff measure, and $\|f\|_{L^\infty (\mathcal H^k)}$ denotes the $L^\infty$ norm of a function $f$ with respect to $\mathcal H^k$.
Let $\Omega$...
- 6,313
5
votes
2
answers
248
views
Hausdorff dimension of the zero set of $\nabla f$
Let $f: \mathbb R^n \to \mathbb R$ be a Lipschitz function with $\nabla f$ nonzero almost everywhere with respect to Lebesgue measure.
What is the supremal Hausdorff dimension of the set on which $f$ ...
- 6,313
4
votes
1
answer
196
views
On the set on which $|Df|$ is maximal for Lipschitz $f$
Let $f: \mathbb R^n \to \mathbb R$ be a Lipschitz continuous function with strict Lipschitz constant $L > 0$.
That is, $|f(x) - f(y)| < L|x - y|$ for all $x \neq y$ in $\mathbb R^d$.
Question: ...
- 6,313
4
votes
1
answer
259
views
Hausdorff dimension of the zero set of the gradient of an eikonal function
Let $f: \mathbb R^n \to \mathbb R$ be a Lipschitz function with $|\nabla f| = 1$ almost everywhere with respect to Lebesgue measure.
What is the supremal Hausdorff dimension of the set on which $f$ is ...
- 6,313