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Questions about geometric properties of sets using measure theoretic techniques; rectifiability of sets and measures, currents, Plateau problem, isoperimetric inequality and related topics.

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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\...
Cosine's user avatar
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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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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 $f$?
Nate River's user avatar
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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 ...
Nate River's user avatar
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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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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^\...
Ribhu's user avatar
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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 ...
No-one's user avatar
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Full disclosure, I posed this question on MSE about a week ago, but realized it may be a better fit here after it sat dormant. For completeness, I'll include the full body of the question here: Let $\...
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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 ...
C. A. Nastasi's user avatar
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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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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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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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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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Note: We denote by $\mu$ be the usual Lebesgue measure on $\mathbb R$. For $E$ a Lebesgue measurable subset of $\mathbb R$, we define its lower asymptotic density at $x \in \mathbb R$ by $$\liminf_{r \...
Nate River's user avatar
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Jerison and Kenig give the following two conditions (alongside an exterior corkscrew condition, which is not relevant to this discussion) in the definition of an $(M,r_0)$ nontangentially accessible ...
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Let $f:\mathbb{R}^2\to \mathbb{R}$ be $L$-Lipschitz. Let $f_\varepsilon:=f*\eta_\varepsilon$ be its smooth $\varepsilon$-mollification, where $\eta_\varepsilon(x)=\frac{1}{C\varepsilon^2}\eta(|x|/\...
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In Maggi's book "Sets of Finite Perimeter and Geometric Variational Problems ", the first variation of perimeter for open sets in $\mathbb{R}^n$ with $C^2$-boundary is given by $ \frac{d}{dt}...
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Let $f: \mathbb R^n \to \mathbb R$ be a $C^{1,1}$ function, i.e. it is continuously differentiable with Lipschitz gradient. For every $\varepsilon > 0$, does there exist a $C^2$ function $g$ such ...
Nate River's user avatar
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Say that a compact $K\subset\mathbb R^d$ has the Wada property if there are disjoint, connected open sets $V_1,V_2,\ldots,V_n\subset\mathbb R^d$, $n\geq3$, such that $K=\partial V_1=\partial V_2=\...
Lavender's user avatar
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Let $E \subset \mathbb{R}^n$ be a set of finite perimeter. Fix $\rho>0$ and a center $x_0\in\mathbb{R}^n$, and write $ B_r := \{x\in\mathbb{R}^n:\ |x-x_0|<r\}, \qquad \partial B_r := \{x:\ |x-...
HIH's user avatar
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Let $M^n$ be a Cartan--Hadamard manifold and $B \subset M$ a geodesic ball. In Kleiner’s proof of the Cartan--Hadamard conjecture in dimension 3, the estimate $$ \max_{\partial E} H_{\partial E} \ge ...
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For $n \geq 2$, let $f: \mathbb R^n \to \mathbb R$ be everywhere differentiable, Lipschitz continuous, and an almost everywhere solution to the eikonal equation $|\nabla f| = 1$ a.e. Question: What is ...
Nate River's user avatar
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Suppose $f: [0, 1] \to \mathbb R$ is continuous and satisfies the following property on its level sets: $$\sum_{t \in \mathbb R} \mu(f^{-1} (t)) = 1,$$ where $\mu$ is the Lebesgue measure, and we ...
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I am a Ph.D. student who is about to begin my dissertation work. So far, I have mainly studied Lawson’s Calibrated Geometries and Leon Simon’s Geometric Measure Theory. At this stage I would like to ...
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Lately, I have spent some time trying to understand the proof of Allard's compactness theorem for integral varifolds. The sources I have been looking at are Section 6 of Allard's original paper (&...
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If we are talking about the Euclidean space $\mathbb{R}^n$, then we may naturally measure what part of the whole space does the Borel set $A$ occupy by simply introducing the notion of the upper ...
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For $0 <\alpha < \frac{1}{3}$, let $C \subset [0, 1]$ denote the fat Cantor set, where intervals of length $\alpha^n$ are removed at every stage. The cumulative distribution function $f: [0, 1] \...
Nate River's user avatar
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Let $f: \mathbb R^n \to \mathbb R$ be Lipschitz and differentiable everywhere. Is it possible that $\nabla f$ is discontinuous almost everywhere?
Nate River's user avatar
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The conformal dimension of a metric space $(X,d)$ is defined as the infimum of the Hausdorff dimensions of all metric spaces quasisymmetric to $(X,d)$. A natural question is whether this infimum is ...
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Let $u:\mathbb{R}^n\to\mathbb{R}$ be a function in $W^{1,2}$ and let $u^*(x)=\lim_{r\to 0} \frac{1}{\omega_n r^n} \int_{B_r(x)} u(y) dy$ be the fine representative of $u$. From Evans-Gariepy Theorem 4....
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It is a (difficult) exercise to show that there exists a measurable set $E \subset [0,1]^2$ (necessarily with zero 2-dimensional Lebesgue measure) such that the projection on every line passing ...
Castoro Moro's user avatar
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Let $V=\underline v(M,\theta)$ be a stationary integral $n$-varifold in $\Bbb{R}^{n+k}$ where $M$ is the $n$-rectifiable set of $V$ and $\theta$ is the multiplicity function. We write $\|V\|=H^n\...
Y.Guo's user avatar
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Let $U \subset \mathbb{R}^n$ be an open domain with smooth (say $C^2$) boundary, and fix a boundary point $p \in \partial U$. Let $T_p(U)$ denote the tangent half-space at $p$, i.e., the blow-up limit ...
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Let $n$ be a positive integer, and $s \leq n$ a positive real number. Does there exist a Lipschitz function $f: \mathbb R^n \to \mathbb R$ such that the set on which $f$ is not differentiable has ...
Nate River's user avatar
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5 votes
1 answer
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Let $n \geq 2$. Does there exist a function $f: \mathbb R^n \to \mathbb R$ that is differentiable everywhere and satisfies $\text{Range}(|\nabla f|) = \{0, 1\}$?
Nate River's user avatar
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12 votes
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Assume that $M$ is a smooth oriented compact manifold with boundary and assume that $\omega$ is a Lipschitz $(n-1)$-form on $M$. Question Is there a published simple proof of the Stokes theorem $$ \...
Piotr Hajlasz's user avatar
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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 $D_0$...
HIH's user avatar
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2 votes
1 answer
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Let $\Omega$ be locally compact topological space. Consider $C_0(\Omega)$ being the space of continuous functions $u$ on $\Omega$ that vanish at infinity, that is, for $r>0$ there is a compacte set ...
Guy Fsone's user avatar
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Let $f: \mathbb R^n \to \mathbb R$ be everywhere continuous, and differentiable with derivative $0$ a.e. with respect to $n-1$ dimensional Hausdorff measure. Is it true that $f$ is necessarily ...
Nate River's user avatar
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I am interested to know whether there are 'optimal' results for convergence in the Hausdorff distance implying some sort of measure convergence? I have not found anything of this site, but I was ...
Keen-ameteur's user avatar
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7 votes
2 answers
644 views

We say a function $f: \mathbb R^n \to \mathbb R$ is essentially continuous if for every $x \in \mathbb R^n$, we have $$\lim_{\delta \to 0_+} \text{esssup}_{y, z \in B_\delta (x)} |f(y) - f(z)| = 0,$$ ...
Nate River's user avatar
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3 votes
2 answers
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This question is a result from my trying to answer this question. I will not repeat that question here, but just formulate the missing piece. Let $W \subset \mathbb{R}^n$ be open and bounded. Suppose ...
Steven's user avatar
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Let $M_1$ and $M_2$ be $n$-dimensional $C^1$ manifolds in $\mathbb{R}^{n+k}$. Let $\mathcal{H}^n$ be the $n$-dimensional Hausdorff measure in $\mathbb{R}^{n+k}$. Is it always true that $\mathcal{H}^n(...
No-one's user avatar
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1 vote
1 answer
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This question is probably quite simple, but I would still appreciate a straightforward answer, whether positive or negative. Thank you in advance. I alredy know that the Besicovitch Covering Theorem ...
Sime's user avatar
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0 votes
1 answer
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Let $ \lambda $ denote the Lebesgue measure on the $ n $-dimensional Euclidean space $ \mathbb{R}^n $. Let $ K \subset \mathbb{R}^n $ be a compact subset whose boundary $ \partial K $ has upper ...
Clement's user avatar
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3 votes
0 answers
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My question is about Besicovich sets in (possibly high) dimension $d\geq 2$, and more precisely about the existing constructions and how small they are. The Kakeya conjecture predicts that they all ...
Mikael de la Salle's user avatar
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3 votes
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I consider the following Brieskorn manifolds for an integer $d\geq 2$: $$ M_d := \left\{(z_1,\ldots,z_d)\in \mathbb{C}^d : |z_1|^2 + \cdots+|z_d|^2=1, z_1^2 + \cdots+ z_{d-1}^2 + z_d^3 =0 \right\}. $$...
Dorian's user avatar
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This question stems from a ZBmath search I did yesterday evening, and it is somewhat related to the following MathOverflow question: "On which regions can Green's theorem not be applied? ". ...
Daniele Tampieri's user avatar
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Let $A\subset\mathbb{R}^n$ be a Borel measurable subset, then a classical result in descriptive set theory says that $A$ is either countable, or contains a Cantor subset $C$ (i.e. a subset ...
simply lemon's user avatar
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1 answer
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Let $$g(x)=\sum_{k=1}^m |P_k(x)|$$ with $P_k$'s the multi-variable polynomials on $\mathbb{R}^n$. Let $Z_g=\{x\in\mathbb{R}^n~|~g(x)=0\}$. Suppose $0\in Z_g$. If $d(\Omega,Z_g):=\inf_{x\in\Omega,~y\in ...
Houa's user avatar
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