# 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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### Prove Property of Doubling Measure on $\mathbb{R}$

Problem. Let $\mu$ be a fixed finite measure on $\mathbb{R}$. $\mu$ is said to be doubling if there exists a constant $C>0$ such that for any two adjacent intervals $I=[x-h,x]$ and $J=[x,x+h]$, ...
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### Hausdorff measure of the middle third Cantor set and Compactness

In the proof of the Hausdorff dimension of the middle third cantor set I cannot understand why we need the following underlined statement. I cannot understand why we need only consider closed ...
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### Doubling measure and Riesz Potential

I am currently trying to solve some analysis exercises on metric spaces, but I cannot quite tackle on of them. The exercises read as follows: Define the measure \mu_x:=\frac{d(\bullet,x)}{\mu(B(x,d(...
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### A formula for Perspective measurement

If I have a photograph of a rectangular object whose image recedes with perspective, is it possible to work out the scale of measurements along that object? For example: I have a photograph of a (...
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### How can a density be larger than $1$?

From Frank Morgan: Geometric Measure Theory, Fourth Edition: A Beginner's Guide, page 13,the $2$-dimensional density of the cone $x^2+y^2=z^2$ at $0$ is $\sqrt{2}$. I feel strange of that，roughly ...
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