  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.
