Questions tagged [geometric-measure-theory]
Questions about geometric properties of sets using measure theoretic techniques; rectifiability of sets and measures, currents, Plateau problem, isoperimetric inequality and related topics.
827 questions
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Does anyone use measures that take values in real numbers and cardinal numbers?
The counting measure on $\mathbb{R}^n$ is a map that takes a subset $A$ of $\mathbb{R}^n$ and returns its cardinality if it is finite or the symbol $\infty$ if it is infinite.
So, if $A\subseteq\...
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Hausdorff dimension of graphs of singular functions
Let $f: \mathbb R^n \to \mathbb R^m$ be continuous, and differentiable almost everywhere with $Df = 0$ almost everywhere.
Question: What is the maximal Hausdorff dimension of the graph of $f$?
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Can a Lipschitz function have derivative 0 on a dense set of small dimension?
Let $f: \mathbb R^n \to \mathbb R$ be Lipschitz continuous. Denote by $Z(f)$ the set on which $f$ is differentiable with derivative $0$.
Suppose $f$ is such that $Z(f)$ is topologically dense.
...
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Hausdorff dimension of the stretch set of a Lipschitz map
Let $f: \mathbb R^n \to \mathbb R^m$ be a non constant Lipschitz map, for $m < n$. We denote by
$$\text{Lip}(f):= \sup_{x, y \in \mathbb R^n} \frac{|f(x) - f(y)|}{|x - y|}$$
the best Lipschitz ...
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Characteristic function of a domain to have higher order variation
For any $k\in\mathbb{N}$, the space $BV^k(\mathbb{R}^n)$ is defined as the set of all functions $f\in L^1(\mathbb{R}^n)$ such that there exists a Radon masure $D^\alpha f$ and for any $\phi\in C^\...
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Approximating the perimeter of a domain
Let $M$ be a Riemannian manifold with the volume measure $\mu$, and $\Omega$ be a bounded open subset of $M$. Assume that $\chi_\Omega$ has bounded variation, that is, $\mathrm{Per}(\Omega)<\infty$....
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Is the restriction of a Sobolev function to some full-measure set continuous?
Motivation: As a consequence of this question, every function $u$ in $W^{1,2}(\mathbb{R}^n,\mathbb{R})$ has a representative $u^*$ such that there is a null set $E\subset \mathbb{R}^n$ with the ...
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Properties of the radial projection of centered convex domains
Suppose that $\Omega_1, \Omega_2 \subseteq \mathbb R^n$ are convex domains.
We assume that they contain the origin. Then the radial projection $P : \partial\Omega_1 \rightarrow \partial\Omega_2$ ...
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Approximation of open set by regular sets
I have the following question: given $\omega\subset \mathbb{R}^d$ a bounded open set and $\eta\in (0,1)$, can I find an open set $\omega_\eta\subset\subset \omega$ with Lipschitz boundary such that $\...
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If Ei converges to E , does its mean curvature converges to those of E?
On a complete, simply-connected Riemannian manifold with nonpositive sectional curvature, assume that every set with $C^{1,1}$ boundary satisfies $\max H \ge c$ for some constant $c$, where $H$ is ...
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$C^{1,1}$ domain plus or minus small ball
In the following link, it says that Lipschitz domain plus or minus small ball may not be a Lipschitz domian.
Therefore, I'm woundering that $C^{1,1}$ domain plus ro minus small ball is a Lipschitz ...
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Lipschitz domain plus or minus small ball
I want to ask a follow up to Intersection between Lipschitz domains.
Let $\Omega\subseteq \mathbb{R}^n$ be a Lipschitz domain with compact boundary. Just to be precise, this means that there are ...
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The jacobian of projection $r_s$ on $C^{1,1}$ surface converges uniformly to 1, when manifold has nonpositive sectional curvature
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 $D_0$...
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Extending Kleiner’s proof of isoperimetric inequality in Cartan-Hadamard manifold to isoperimetric regions with nonsmooth boundary
Let $M^n$ be a Cartan--Hadamard manifold and $B \subset M$ a geodesic ball. In Kleiner’s proof of the Cartan--Hadamard conjecture in dimension 3, the estimate
$$
\max_{\partial E} H_{\partial E} \ge ...
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Are these two definitions of an NTA (nontangentially accessible) domain equivalent? If so, is the constant unchanged?
Jerison and Kenig give the following two conditions (alongside an exterior corkscrew condition, which is not relevant to this discussion) in the definition of an $(M,r_0)$ nontangentially accessible ...
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