**Research Project Title:**

### Isoperimetric Profile and Medial Axis Curves

**abstract:**When studying districting patterns like gerrymandering, there is a variety of measures we can use to quantify the compactness of a shape. The most commonly used measure is the isoperimetric quotient, the ratio of a shape’s area to its perimeter. However, for a given shape, we can perturb the perimeter enough to drastically alter this ratio without affecting the shape’s larger features. In order to better quantify compactness, we need a measure that captures information about a shape at multiple scales. The isoperimetric profile, which is defined as the largest area enclosed by a given perimeter within the shape is much more resistant to small perturbations. While some work has been done to calculate the convex envelope of isoperimetric profile, there is no known algorithm to compute the exact profile, even for simple, two dimensional polygons. The goal of my project is to devise such an algorithm.

**About:**
"I’m interested in participating in SuperUROP because I would like to get some more research experience. I’m excited about the project because I particularly enjoy numerical algorithms."