UBC CORDS/COCANA Operations Research Seminar: Serhii Myroshnychenko

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.