Where Does the Centroid Go? Balancing Shadows and Slices of Convex Bodies

Abstract: The centroid is a fundamental notion of center: it represents the center of mass in mechanics, the mean in probability and statistics, and a basic descriptor in data analysis and imaging. This talk asks what happens to the centroid when a high-dimensional convex body is viewed through a lower-dimensional slice or shadow. Surprisingly, neither operation generally preserves it. We present three sharp results. Sections through the centroid satisfy an optimal bound on their imbalance; the projection of a centroid and the centroid of the projection differ by at most about 20.2% of the relevant width; and, in sufficiently high dimensions, a body may have only one central hyperplane section sharing its centroid. The emphasis will be on the geometric ideas behind these results, including first moments, concavity, symmetrization, and spherical transforms.

Mathematics

Audience: researchers in the topic

PIMS-CORDS SFU Operations Research Seminar