# 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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### Can objects with different numbers of dimensions be compared?

To elaborate on the title: In many places, I've seen people claiming that objects residing in different dimensional spaces are not commensurable; that is, there is no way to state whether a cube of ...
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### Can positivity of currents implies positivity of forms?

Let $\alpha$ and $\beta$ be 2 continuous (or smooth) forms of $(1,1)$-type on a complex manifold $X$. Of course they can be considered as currents. Assume $\alpha\geq \beta$ in the sense of currents. ...
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### Inequality of Hausdorff measures for convex sets $\mathfrak{H}^{n-1}(\partial E)\le \mathfrak{H}^{n-1}(\partial F)$

I'm preparing for an exam on the calculus of variations and I need help in solving this exercise from an old exam text (actually it's only a part of a bigger exercise but its parts are quite ...
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### Isoperimetric inequality for bounded domains in $\mathbb{R}^n$

I'm looking for the proof of the following/reference to such proof. At the end of the day, my goal is to confirm if this inequality holds. Let $H_{n-1}$ be the $n-1$ dimensional Hausdorff measure ...
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I found, with no reference, the following theorem which is called Besicovitch derivation theorem. Do you know any article/book where I can find this powerful result? Let $\Omega\subset \mathbb R^n$ ...
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