Convexity Properties of the Condition Number

In the space of n × m matrices of rank n, n ≤ m, consider the “condition metric”, obtained by multiplying the usual Frobenius Hermitian product by the inverse of the square of the smallest singular value. We prove that this last quantity is logarithmically convex along geodesics in that space. Let N be a complete submanifold of R and let R be endowed with the analogous “condition metric”, obtained by multiplying the usual metric by the square of the inverse of the distance to N . We prove that the distance to N is logarithmically convex along geodesics in that space. Mathematics Subject Classification (MSC2000): 65F35 (Primary), 15A12 (Secondary). C. Beltran, M. Shub, Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4 ([email protected]), ([email protected]). CB was supported by MTM2004-01167 and by a Spanish postdoctoral grant. CB and MS were supported by an NSERC Discovery Grant. J.-P. Dedieu, Institut de Mathématiques, Université Paul Sabatier, 31062 Toulouse cedex 09, France ([email protected]). J.-P. Dedieu was supported by the ANR Gecko. G. Malajovich, Departamento de Matemática Aplicada, Universidade Federal de Rio de Janeiro, Caixa Postal 68530, CEP 21945-970, Rio de Janeiro, RJ, Brazil ([email protected]). GM is partially supported by CNPq grants 303565/2007-1 and 470031/2007-7, by FAPERJ (Fundação Carlos Chagas de Amparo à Pesquisa do Estado do Rio de Janeiro) and by the Brazil-France agreement of cooperation in Mathematics.