Sorry for asking such a basic question, but this is not my area of expertise. In my work I'm using the coarea formula: for $\Omega \subseteq \mathbb{R}^n$ open and bounded, and $u …

Why do currents, functionals on compactly supported differentiable n-forms, bear the name they do? I've assumed that it has something to do with an electrical current being formal …

A metric space is doubling if any ball of radius $2R$ can be covered by $N$ balls of radius $R$ and $N$ is fixed once forever. Is there an example of complete length-metric spa …

Let $A\subset \mathbb{R}^n$ be an open bounded set that is the interior of a closed set. Does there exist a sequence of open sets $A_j\subset \subset A$ such that the (Lebesgue) me …

Hi ; i have this theorem from the book :Set-valued analysis Let $(\Omega,\mathcal{A},\mu)$ be a complete $\sigma$-finite measure space , $X$ a complete separable metric …

Let $X$ be a metric space, $\Sigma_{1}$ the borel sigma algebra and $\Sigma_{2}$ the sigma algebra generated by balls (open and closed). If $\mu$ is a probability measure on $\Si …

There is a measure preserving map from the unit interval onto the unit cube that is Lipschitz of order 1/2, that is $|f(x)-f(y)| \leq A |x-y|^{1/2}$. By considering the image of sm …

Hello: This is my first time posting on mathoverflow. It is a fairly difficult question. I've posted it on Math Stack Exchange, and got one upvote, no comments and no answers. …

I'd like to capture the intuitive notion that a Jordan curve $\gamma_2$ “follows” or “approximates” another Jordan curve $\gamma_1$, i.e. goes somehow &ldq …

I wanto know the smoothness of convex set in ${\bf R}^n$. Recall the following definition. Definition : $X$ is a bounded closed convex set in ${\bf R}^n$ if for $x$, $y\in X$, …

For a mapping $f: \Omega\to \bf{R}^n$, what kind of condition ensures that the one-dimensional Hausdorff measure of $f^{-1}(E)$ is zero whenever $E$ is of zero one-dimensional Haus …

Let $S$ be a subset of $\mathbb{R}^2$ with the following property. For all $x \in \mathbb{R}^2$ and $\varepsilon \gt 0$, there exists a nontrivial interval $[a,b] \subseteq [1-\va …

Let $A\subset \mathbb{R}^n$ be a (measurable) bounded set, and consider the following optimization problem: minimize $P(X)$, the perimeter of a set $X$, where $X$ ranges over all C …

Let S be a n-rectifiable subset of \mathbb{R}^N , we define the differentiability of a funtion f:S \to \mathbb{R} at a point x_0 in S as in Federer's book, where he called differen …

While not so well known, the von Neumann paradox is built among the same lines, in dimension 2 and with transforms within the special linear group. But what is wrong with the foll …