Geometric Measure Theory

How to add shortcuts in the plane Imagine you have a network of roads. One way to judge the efficiency of the roads is in terms of how much longer it takes to travel between two points in your network than if you could just drive off-road in a straight line. This measure of efficiency is called quasiconvexity. Definition: A set $E$ is $C$-quasiconvex meaning that for every $x,y\in \tilde{\Gamma}$, there is a curve $\gamma\subseteq \mathbb{R}^{2}$ connecting $x$ and $y$ so that $\ell(\gamma)\leq C|x-y|$. Not every connected set of finite length is quasiconvex, and this note is about the problem of adding shortcuts to make it so.
7 min read
We discuss an open problem about parametrising boundaries of semi-uniform domains, a class of domains introduced by Aikawa and Hirata. The question is essentially whether one can generalise the result that curves of bounded turning in the plane are quasi symmetric images of the real line, or they are bi-Lipschitz images if additionally they are Ahlfors regular. We first give a general background of the problem.
11 min read
In Week 5 of my Geometric Measure Theory course, we’ll be studying Lipschitz functions and various useful techniques for working with them. In this note I’m going to discuss one of my favorite results from this week and show how I illustrated it using Numpy and Matplotlib. Neither the theorem nor the code are difficult, but I’ve always wanted to visualize this theorem, and coding it is a nice exercise.
7 min read
In Week 3 of the Geometric Measure Theory course I’m teaching, an important class of objects we discuss are Frostman measures. An s-Frostman Measure is a measure $ \mu$ in Euclidean space so that $ \mu(B(x,r))\leq Cr^s$ for any ball $ B(x,r)$. In this note I show how I generated illustrations of them using Matplotlib.
5 min read