# 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 and 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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### Extending the Homotopy formula of Federer 4.1.9 to Riemannian Manifolds

Is it possible to extend the Homotopy Formula expressed for classical currents in open sets of $\mathbb{R}^n$ to currents in a Riemannian manifold? (see Federer's book Geometric Measure Theory 4.1.9) ...
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### Box counting dimension of the graph of the Cantor function

Consider the Cantor staircase function. The Hausdorff dimension of its graph is $1$. What is the box-counting dimension of its graph? A more general question is on MathOverflow.
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### Set of positive reach and Minkowski sum

Let us denote by $A\subset \mathbb{R}^2$ a set with positive reach and suppose $\rm{reach}(A) \ge r$. It is true then that the Minkowski sum $A\oplus B_r$ has regular boundary (say piecewise Lipschitz)...
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### Hausdorff measure independent of metric

I have seen in several papers the claim that for a compact Riemannian manifold the Hausdorff measure will be independent of the Riemannian metric chosen. Could someone explain what this means ...
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### Definition of $\ell$-dimensional points [Geometric Measure Theory]

Reference: Cheeger, J., Colding, T., _On the structure of spaces with Ricci curvature bounded below., J. Differential Geometry, 45 (1997) 406--480. I have a question regarding the following ...
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### lower semi-continuous map to a probability measure

Let $\mu$ be a probability measure and $R(p)$ an interval with the property that both $\inf(R(p)):=\underline{r}(p)$ and $\sup(R(p)):=\bar{r}(p)$ are continuous and bounded functions of $p\in [0,1]$, ...
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### Is this function continuous/smooth?

Suppose $A \subset \mathbb{T}_2$ is a measurable set, where $\mathbb{T}_2$ is the torus group. For a fixed $n$, define the function $f_A:\mathbb{T}_2^n \rightarrow \mathbb{R}$ as f_A(x_1, ..., ...
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### Best topics to study in order to research geometric measure theory

What are the best topics (other than GMT itself) to learn in order to pursue research in the field of geometric measure theory?