Convexity properties of the condition number II

In our previous paper [SIMAX 31 n.3 1491-1506(2010)], we studied the condition metric in the space of maximal rank matrices. Here, we show that this condition metric induces a Lipschitz-Riemann structure on that space. After investigating geodesics in such a nonsmooth structure, we show that the inverse of the smallest singular value of a matrix is a log-convex function along geodesics (Theorem 1). We also show that a similar result holds for the solution variety of linear systems (Theorem 31). Some of our intermediate results, such as Theorem 12, on the second covariant derivative or Hessian of a function with symmetries on a manifold, and Theorem 29 on piecewise self-convex functions, are of independent interest. Those results were motivated by our investigations on the com- plexity of path-following algorithms for solving polynomial systems.