Questions tagged [geometric-measure-theory]

Question 1

Let $f:M\to\mathbb{R}^k$ be smooth, where $M$ is a closed manifold of dimension $n>k$. I am thinking about the continuity properties of the map $c\mapsto\mathcal{H}^{n-k}(f^{-1}(c))$. Clearly this ...

Question 2

Let $\Omega \subset \mathbb R^d$ be a bounded set with sufficiently nice boundary and let $$\mathbb I _\Omega(x):= \begin{cases}1 &x\in \Omega,\\ 0& \text{else}\end{cases}$$ be its indicator ...

Question 3

I was reading Francesco Maggi's Sets of Finite Perimeter and Geometric Variational Problems which states that the Hausdorff measure $\mathcal{H}^s$ is Borel regular for $s >0$. My question is, what ...

Question 4

A set $E \subset \mathbb{R}^d$ is called $1$-rectifiable if there exists a (countable) family of Lipschitz mappings $f_i : \mathbb{R} \rightarrow \mathbb{R}^d$ such that $$ H^1 \bigg(E \setminus \...

Question 5

I have a question about Hausdorff outer measures (basically the title), $A \subset B \implies \mathcal{H}_\delta^p(A) \leq \mathcal{H}_\delta^p(B)$, but $\delta_1 < \delta_2 \implies \mathcal{H}^p_{...

Question 6

$K \subset \mathbb{R}^n$ is any compact set. Does its $\epsilon$-neighborhood $K_{\epsilon} := \{x\in \mathbb{R}^n : \text{dist}(x,K) < \epsilon\}$ have Lipschitz boundary? (Do we have to require $\...

Question 7

I'm currently working through Leon Simon's book on GMT, and have come to problem 2.6 which I reproduce below. I'm happy with the first part, but the second part worries me because I seem to have ...

Question 8

Moser's Worm Problem asks what is the minimum area of a region that can cover every plane curve of length $1$. I see, on Wikipedia, that there is a known lower bound for area of this region, if we ...

Question 9

In section 7.1(page 98) of "The Geometry of Fractal Sets" by Falconer he introduces the Kakeya problem and a related problem, solved by Besicovitch, which is the following: "If $f$ is a ...

Question 10

Let (X,d) be a complete, locally compact metric space, n-countably rectifiable for a positive integer n. For $x\in X$, denote by $$ B(x,r):=\{y\in X|d(x,y)<r\} $$ Let $\mu$ be a non-negative ...

Question 11

Let $M$ be a 3-dimensional Cartan–Hadamard manifold (complete, simply connected, nonpositive sectional curvature). Suppose $S \subset M$ is a $C^{1,1}$ surface enclosing a domain $E$, and let $D_0$ be ...

Question 12

Let $\tilde Q_d$ denote the following analogue of the middle-thirds Cantor set in $\mathbb R^d$: start with a $d$-dimensional hypercube, and at the $n$th stage of the construction remove the middle ...

Question 13

Let $M$ be a $3$--dimensional Cartan--Hadamard manifold (complete, simply connected, nonpositive sectional curvature). Suppose $S\subset M$ is a $C^{1,1}$ surface which encloses a domain $E$. Let $D_0$...

Question 14

Let $M$ be a $3$--dimensional Cartan--Hadamard manifold (complete, simply connected, nonpositive sectional curvature). Suppose $S\subset M$ is a $C^{1,1}$ surface which encloses a domain $E$. Let $D_0$...

Question 15

Let $M$ be a $3$--dimensional Cartan--Hadamard manifold (complete, simply connected, nonpositive sectional curvature). Suppose $S\subset M$ is a $C^{1,1}$ surface which encloses a domain $E$. Let $D_0$...

Stack Exchange Network

Questions tagged [geometric-measure-theory]

Smooth functions with arbitrarily large regular level sets

What are the higher order (distributional) gradients of an indicator function?

Why is the Hausdorff measure $\mathcal{H}^s$ not Borel regular for $s =0$?

Rectifiable Subsets of Sets with Large Hausdorff Dimension

$\mathcal{H}_\delta^p(A) \leq \mathcal{H}_\delta^p(B)$, but $\delta < \delta_1 \implies \mathcal{H}^p_{\delta_1} \leq \mathcal{H}^p_{\delta}$?

Does the $\epsilon$-neighborhood of a compact set have Lipschitz boundary?

A potential counterexample to a problem in Simon's GMT book

Minimum area for the non-convex case of Mosers Worm Problem?

Kakeya problem and Riemann integral

$\mu^s(S)\ge \int_S \lim\limits_{r\to 0}\mu^s(B(x,r)) dH^k$ for a non-negative radon measure $\mu^s$ in the following case?

How to prove the following inequality in Cartan–Hadamard manifold

Intersection of a fractal with a sphere

Continuity of tangent vectors to normal geodesics in Cartan–Hadamard manifolds

How to derive $ \frac{d}{dt} H(t)\;=\; -\big(\operatorname{Ric}(N,N)+|\sigma|^2\big), $ using Riccati equation

The jacobian of projection $r_s$ on $C^{1,1}$ surface converges uniformly to 1, when manifold has nonpositive sectional curvature

Hot Network Questions