Quantitative differentiation and its applications

In a beginning calculus course, one of the many concepts we learn about is differentiability. Recall that informally we say a function 𝑓 is differentiable at a given point 𝑥 if there exists an affine function that looks nearly identical to 𝑓 on an infinitesimal scale near 𝑥. In this thesis, we will be discussing a similar concept called “quantitative differentiation”, clarifying and further explaining notes of Robert Young. The word “quantitative” means that instead of talking about differentiability at a point we talk about differentiability on a large scale. We will also discuss extensions of this concept for functions on higher dimensional domains and into arbitrary metric spaces. To quantify how often a function is quantitatively differentiable, we will use the notion of a “Carleson set” of locations and scales. After this we will discuss applications. One specific application we will discuss is the concept of the “medial axis” of a shape: the set of all points having more than one closest point on the shape. This is an important object of study in both pure mathematics and in image processing.