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# 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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### Why is studying upper bounds for $|I_\delta(\mathcal P,\mathcal L)|$ useful?

A natural problem in incidence geometry is counting the number of incidences of points and lines. For example, if $\mathcal P$ is a collection of points in $\Bbb R^d$, and $\mathcal L$ is a collection ...
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### Integral inequalities with total variation measure

I know and I can prove that, given $f:\Omega \rightarrow \mathbb{R}$ in $L^1_{|\mu|}(\Omega)$, then $$\left|\int_{\Omega}f\,d\mu\right|\leq \int_{\Omega}|f|d|\mu|$$ where $|\mu|$ is the total ...
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### When does a random geometric graph become connected?

Fix $n\in \mathbb N$ and let $X_1,\dots,X_n$ be i.i.d uniform random points in $[0,1]^2$. For $r\in \mathbb R$ consider the (random) geometric graph $\mathcal G _r(X)$ with vertices $X=\{X_i\}$ and ...
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### A question on the Hausdorff dimension of a subset of $\mathbb{R}.$ [closed]

Let $p\in [0,1].$ I am interested in showing that there exist sets $A,B\subset \mathbb{R}$ of Hausdorff dimension $p$ such that the $p$-dimensional Hausdorff measures $H_p(A)=\infty$ and $H_p(B)=0$. I'...
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