DIMACS - RUTGERS EXPERIMENTAL MATHEMATICS SEMINAR

Sponsored by the Rutgers University Department of Mathematics and the
Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)

Co-organizers:

Drew Sills, Rutgers University, asills {at} math [dot] rutgers [dot] edu

Doron Zeilberger, Rutgers University, zeilberg {at} math [dot] rutgers [dot] edu

Title: A generalization of the Chandler Davis convexity theorem and applications

Speaker: Yury Grabovsky, Temple University

Date: December 8, 2005 5:00pm

Location: Hill Center, Room 705, Rutgers University, Busch Campus, Piscataway, NJ

Abstract:

The classical theorem of Chandler Davis says that a rotationally invariant function on symmetric matrices is convex if and only if it is convex on the diagonal matrices. In this paper we uncover a simple abstract mechanism behind this classical theorem. A motivation for the present work comes from applied mathematics. In a problem from the mathematical theory of composite materials it was important to understand how to construct smallest convex and rotationally invariant sets containing a given set. The new twist was that the action of the rotation group and convexity happen in different coordinate systems. Equivalently, we may consider a problem in a single coordinate system but with a non-linear group action. The combination of convexity and rotational invariance drew us to Davis's theorem, which we generalize for non-linear group actions and infinite dimensional vector spaces.