New answers tagged geometric-measure-theory
5
votes
Accepted
Macroscopic sets - a notion of largeness for Lebesgue null sets
By Frostman's lemma, if $E$ is a compact set of positive $\alpha$-Hausdorff content, then there exists a probability Borel measure $\mu$ supported in $E$ such that $\mu(I) \leq c |I|^\alpha $ for ...
2
votes
Accepted
Computation of tangent functional
$\newcommand{\om}{\omega}\newcommand{\Om}{\Omega} \newcommand{\de}{\delta}\renewcommand{\th}{\theta}$For each real $t>0$ and some $\om_t\in\Om$,
\begin{equation*}
\|x+ty\|
=|(x+ty)(\om_t)|=|...
- 110k
2
votes
Computation of tangent functional
The idea is that, for $|x(\omega)|\sim 1$ and $t$ small, $$|x(\omega)+ty(\omega)|\sim|x(\omega)^2+tx(\omega)y(\omega)|\sim 1+tx(\omega)y(\omega)$$
Indeed, if $|x(\omega)|<1-\varepsilon$, then for ...
- 354
2
votes
Accepted
Gateaux differentiability of the norm in Banach spaces
We have the following definitions:
\begin{equation*}
V^*:=\{f\in S^*\colon\exists x\in S\ f(x)=1\},
\end{equation*}
where $S$ and $S^*$ are the unit spheres in the underlying Banach space $E$ and ...
- 110k
2
votes
Accepted
Potentially elementary question on affine functions on Banach spaces
A function $f$ from a linear (or, more generally, affine) space $A$ is affine (see e.g. this post) if for all $a$ and $b$ in $A$ and all real $t$ one has $f((1-t)a+tb)=(1-t)f(a)+tf(b)$. If $f$ is ...
- 110k
2
votes
Accepted
Is every area-minimizing cone a level set of a least-gradient function?
Yes. Let $L := \mathbf C \cap \partial B_1$ be the link of $\mathbf C$. Since $\mathbf C$ meets $\partial B_1$ transversely and is smooth near $\partial B_1$, $L$ can be viewed as a closed submanifold ...
- 430
6
votes
What is an "open Baire set"?
The function $\varphi$ in H.-J. is also assumed to be lower semi-continous, which is why $\{a<\varphi\}$ is open. This set is a Baire set because $\varphi$ is a Baire function.
- 1,714
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