Prerequisites: multivariable calculus, functional analysis, measure theory, and differential geometry.
This project explores a facet of complex geometry that can be approached using convex geometry. An n-dimensional convex body is defined as a closed convex subset of n-dimensional Euclidean space with non-zero volume. The classical Hausdorff distance establishes a metric on the space of all convex bodies, satisfying the Borel-Heine property via the Blaschke selection theorem. A natural metric on the space of all convex bodies can also be derived using mixed volumes, inspired by a construction from complex geometry. The goal of this REU is to compare these two metrics and their resulting topologies. Through the lens of toric symmetries, the results aim to shed light on the metric geometry of singularity types, as introduced by Darvas-Di Nezza-Lu.