# 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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### Geometrical interpretation of complex exponential integral

Coefficients of Fourier series of a function $f$ are computed by multiplying $f(x)$ by the exponential term $e^{-inx}$, then by integrating $f(x)e^{-inx}$ from $-\pi$ to $\pi$ and dividing by $2\pi$ (...
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### $f : \mathbb{R} \to \mathbb{R}$ (Lipschitz) continuous implies $f(A)$ is Borel for all Borel $A$.

Full question: Let $(\mathbb{R}, \mathfrak{M}, m)$ denote the measure space $\mathbb{R}$ equipped with the Borel $\sigma$-algebra and the Lebesgue measure. Suppose $f : \mathbb{R} \to \mathbb{R}$ is ...
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### Uncountable sets of Hausdorff dimension zero

Let $A \subset \mathbb{R}$ be a countable set. It is easy to see that $A$ has Hausdorff dimension $\dim_H(A) = 0$. Do there exist uncountable sets $A \subset \mathbb{R}$ with $\dim_H(A) = 0$?
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### Polar coordinates, bounded domain with $C^{1}$ boundary

I have a question about a integral on a surface. It is well known that for any Integrable function $f$ defined on $\mathbb{R}^{n}$, it holds that (1) \quad \frac{d}{dr} \int_{B(0,r)}...
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### Minimum of $F$ over Finite Perimeter Sets in $\mathbb R^N$

Problem: Let $G$ be a bounded Borel set. Let $X$ be the set of finite perimeter sets in $\mathbb R^N$ and $F: X \to \mathbb R \cup \{+\infty\}$ defined as \[ F(E)= \begin{cases} Per(E) \hspace{1,...
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### Surface area from indicator function

I know that the volume and the surface area of a sphere of radius $R$ are related by a derivative: $$V(R)=\frac{4}{3}\pi R^3$$ $$A(R)=4\pi R^2=\frac{\partial V(R)}{\partial R}$$ I am asking if an ...
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### How difficult is it to impose a differential structure on a fractal?

Assuming we have a 4-dimensional smooth manifold $M$ embedded in $\mathbb{R}^{m}$ that is difficult to understand its differential structure. For convenience's sake we can assume it is compact, simply ...
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### Let $f: [0,1] \to \mathbb R$ and $\Gamma(f)$ its graphic. Show that $\mathcal H^1(\Gamma(f)) \geq 1$.

Problem: Let $f: [0,1] \to \mathbb R$ and $\Gamma(f)$ its graphic. Show that $\mathcal H^1(\Gamma(f)) \geq 1$. Attempt: Well, if $f \equiv 0$ we get 1. Provided some sort of goodness like $f \in C^1$ ...
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### Uniform random samples inside bounded region

In an $n$ dimensional space I have a region bounded by pairs of hyperplanes: $$b_j \le \sum_{i=1}^n a_{ij} x_i \le c_j, \quad\forall j=1,\ldots,m.$$ We can include in those ...
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### Use uniform convergence to control the derivative of function

Let $f_n$: $I:=[-1,1]\to \mathbb R$ be given, and we also assume that $f_n\in C^\infty(I)$. Assume $f_n\to 0$ uniformly in $I$, and we know $$\limsup_{n\to\infty}\int_I\sqrt{1+(f'_n)^2}<\infty$$ ...
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### Generalizing Green's Theorem

Let $\phi:[0,1] \rightarrow \mathbb{R}^2$, with $\phi(t)=(x(t),y(t))$, a function satisfying the following assumptions: (i) $x(t)$ and $y(t)$ are absolutely continuous; (ii) $\phi(0)=\phi(1)$, the ...
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### Pointwise convergence of a sequence of approximate limits of BV functions.

So, let $\Omega\subset \mathbb R^2$ bounded and consider a sequence of functions $\{u_k\}_{k\in\mathbb N}\subset BV(\Omega)$ and $u\in BV(\Omega)$ such that $u_k\rightarrow u$ weakly* in $BV(\Omega)$. ...
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### Example of a compact non-$G_{\delta}$ set?
For a homework assignment, I recently solved the following problem in Folland's real analysis text. Folland's Question: If $X$ is a Locally Compact Hausdorff space, and $K \subset X$ is a compact \$...