Geodesic diameter of sets defined by few quadratic equations and inequalities

摘要

We prove a bound for the geodesic diameter of a subset of the unit ball in ? ~n described by a fixed number of quadratic equations and inequalities, which is polynomial in n, whereas the known bound for general degree is exponential in n. Our proof uses methods borrowed from D'Acunto and Kurdyka (to deal with the geodesic diameter) and from Barvinok (to take advantage of the quadratic nature).