Convexity Properties of the Condition Number II

In our previous paper [1], we studied the condition metric in the space of non-singular n ×m matrices. Here, we show that this condition metric induces a Lipschitz-Riemann structure on the space of all n×m matrices. 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). †Mathematics Subject Classification (MSC2010): 53C23 (Primary), 65F35, 15A12 (Secondary). ‡C. Beltrán, Departamento de Matemáticas, Estad. y Comput. Universidad de Cantabria, Santander, España (carlos.beltran@unican.es). CB was supported by MTM2007-62799 and MTM2010-16051, Spanish Government. §J.-P. Dedieu, Institut de Mathématiques, Université Paul Sabatier, 31062 Toulouse cedex 09, France (jean-pierre.dedieu@math.univ-toulouse.fr). J.-P. Dedieu was supported by the ANR Gecko and by the Fields Institute at Toronto. ¶G. Malajovich, Departamento de Matemática Aplicada, Universidade Federal de Rio de Janeiro, Caixa Postal 68530, CEP 21945-970, Rio de Janeiro, RJ, Brazil (gregorio.malajovich@gmail.com). He was partially supported by CNPq, FAPERJ and CAPES from Brazil, and by the Brazil-France agreement of cooperation in Mathematics. ‖M. Shub, CONICET, IMAS, Universidad de Buenos Aires, Argentina and CUNY Graduate School, New York, NY, USA. (shub.michael@gmail.com) M.S. was partially supported by a NSERC Discovery Grant, and by a CONICET grant. ∗∗J.-P.D., G.M. and M.S were partially supported by the MathAmSud grant Complexity. 1 ar X iv :0 91 0. 59 36 v2 [ m at h. D G ] 1 6 Se p 20 11 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 complexity of path-following algorithms for solving polynomial systems.

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