# Tagged Questions

The study of the geometric structure of measures, as well as the study of geometry from a measure-theoretic viewpoint, geometric measure theory has applications in partial differential equations, harmonic analysis, differential and Riemannian geometry, as well as calculus of variations. Statements ...

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### To show that the set point distant by 1 of a compact set has Lebesgue measure $0$

Could any one tell me how to solve this one? Let $K$ be a compact subset of $\mathbb{R}^n$, and $$A:=\{x\in\mathbb{R}^n:d(x,K)=1\}.$$ Show that $A$ has Lebesgue measure $0$. Thank you!
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### Hausdorff Dimension of Set of Measure Zero

It's clear that every $A \subset \mathbb R^n$ with $\dim_H(A) < n$ we have $\mathcal H^n(A) = 0$. Is there any $A \subset \mathbb R^n$ with $\mathcal H^n(A) = 0$ but $\dim_H(A) = n$? Thank you.
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### Higher Order Coarea Formula

I was wondering, if there is a generalization of the coarea formula to higher order derivatives, which would allow one, for example, to relate the norm of the Hessian of a real-valued function $u$ to ...
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### Fourier dimension of a measure restricted to an open set

Suppose that the measure $\mu$ on $\mathbb{R}^n$ has Fourier dimension $\beta$, which is to say that \begin{equation*} \beta= \sup\left\{\gamma \leq n : |\hat{\mu}(x)| \leq C(1+|x|)^{-\gamma/2}\right\...
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### Is the validity of measuring area by approximation an assumption of calculus?

The assumption that if you subdivide an area into more and more sub intervals, the approximation gets better and better. Has this been formally proved, or is it just intuition? Thanks!
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### Calculating the upper Minkowski dimension of the set $\{0,1,\frac{1}{2}, \frac{1}{3}, \ldots \}$

The upper Minkowski dimension of a compact set $A$ in $\mathbb{R}$ is defined as  \overline{\dim}_M = \inf \{ \epsilon > 0 : \text{ there is a constant } C(\epsilon) \text{ such that } \mu(A_\...
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### Volume is a Continuous Function

I am working on the following problem: Suppose $C \subset \mathbb{R}^d$ is a compact and non-empty set. Let $C_0 = C$ and let $C_t = \{x \in \mathbb{R}^d : d(x,C) \leq t \}$ for all $t >0$. ...
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### Any two polish atomless probability spaces are isomorphic

On page 7 of Villani's Optimal Transport: Old and New (page 19 in this preprint), he states that any two polish atomless probability spaces $(\mathcal{X},\mu)$ and $(\mathcal{Y},\nu)$ are measure-...
### Lipschitz continuous one-to-one mapping from subset $K\subset\mathbb{R}^n$ of positive measure to $\mathbb{R}^{n-1}$
Let $f:\mathbb{R}^n\to \mathbb{R}^{n-1}$ and $K\subseteq \mathbb{R}^n$ be a set of positive Lebesgue measure. What kind of regularity do we have to impose on $f$ (e.g., $C^1$, Lipschitz) to conclude ...