# Questions tagged [geometric-measure-theory]

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 geometry, Riemannian geomerty, sub-Riemannian geometry, as well as calculus of variations. Statements such as the isoperimetric inequality and the coarea formula, and subjects such as the Plateau problem belong under this tag.

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### Convexity of a set of probability densities

Let $Q =$ {$d$-dimensional probability densities with independent marginals}, i.e. $$Q = \left\{ q(x) = \prod_{i=1}^d q_i(x_i)\;\; \big| \;\; x\in \mathbb{R}^d\right\}.$$ I'm wondering if this is ...
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### Pointed Measured Gromov-Hausdorff Convergence

It is well-known that one can extend the definition of Gromov-Hausdorff convergence to non-compact metric spaces by instead considering pointed Gromov-Hausdorff convergence. There is also an ...
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### A generalization of Green-Gauss divergence theorem to Sobolev functions on sets of finite perimeter

Let $n\ge 2$. Let $u \in W^{1,1}(\mathbb{R}^n)$, i.e. $u \in L^1(\mathbb{R}^n)$ and its distributional gradient is represented by an element of $L^1(\mathbb{R}^n;\mathbb{R}^n)$. By the Sobolev ...
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### Measurability of Hausdorff measure under bi-Lipschitz mappings

For $0< s < n$, denote by $\mathscr{H}^s$ the $s$-dimensional Hausdorff measure. Let $E\subseteq \mathbb{R}^n$ be $\mathscr{H}^s$-measurable, and $f:\mathbb{R}^n \to \mathbb{R}^n$ a bi-Lipschitz ...
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### Geometric condition number for uniform sampling a real algebraic set

A problem I'm on a hunt to figure out. I'm not sure whether it's better placed here or on Overflow- it has the feeling of being very standard for the right person but I haven't done enough geometric ...
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1answer
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### Computing $n$-dimensional Lebesgue measure with Euclidean $n$-balls

I am studying the coarea formula proof from Evans and Gariepy's Measure Theory and Fine Properties of Functions. At the start of lemma 3.5, the authors are assuming that we can compute the $n$-...
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### Strong convergence vs Weak convergence _ compactness of integral varifolds

I am reading the proof of the compactness of Integral varifolds on L.Simon's book " Lecture on geometric measure theory", there is a part of the proof concerning the conclusion that the ...
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### Is the Hausdorff measure absolutely continuous with respect to the lebesgue measure? $H^{d-1} \ll \lambda^d$

I'm sure this is just basic theory but I can't find this ANYWHERE. Is it true that $$H^{d-1} \ll \lambda^d$$ where $H^{d-1}$ is the $(d-1)$-dimensinal Hausdorff measure and $\lambda^d$ is the $d$-...
1answer
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### What's the intuition behind the Co-Area formula?

I mainly work in statistics and I know only basic measure theory. I was trying to understand the Co-Area formula by Federer. If $f:\mathbb{R}^M\to \mathbb{R}^N$ is a Lipschitz function with $M \geq N$...
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### Not Jordan measurable

How to prove $\forall Q \cap [0,1] \times [0,1]$ is not Jordan measurable? I know as the only rectangle contain by this are points so the inner will be 0, but what about the outter? Is there any way ...
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0answers
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