Math Seminars.
Alexander Barvinok (U of M),
Quadratic Convexity.
Let f: R^n ---> R^2 be a map given by a pair of quadratic
forms. A (strong) version of the Toeplitz-Hausdorff Theorem asserts
that the image f(S) of the unit sphere S in R^n is a convex set in R^2
provided n>2. A similar result holds for the map C^n ---> R^3 defined by
3 Hermitian forms. In this talk, I discuss convexity properties of the
image for more than 2 real (3 Hermitian) forms. New convexity results
will be presented as well as connections with distance geometry and
molecular conformation.