We define in the space of $n\times m$ matrices of rank $n$, $n\leq m$, the condition Riemannian structure as follows: For a given matrix $A$ the tangent space at $A$ is equipped with the Hermitian inner product obtained by multiplying the usual Frobenius inner product by the inverse of the square of the smallest singular value of $A$ denoted $\sigma_n(A)$. When this smallest singular value has multiplicity 1, the function $A\rightarrow\log(\sigma_n(A)^{-2})$ is a convex function with respect to the condition Riemannian structure that is $t\rightarrow\log(\sigma_n(A(t))^{-2})$ is convex, in the usual sense for any geodesic $A(t)$. In a more abstract setting, a function $\alpha$ defined on a Riemannian manifold $(\mathcal{M},\langle,\rangle)$ is said to be self-convex when $\log\alpha(\gamma(t))$ is convex for any geodesic in $(\mathcal{M},\alpha\,\langle,\rangle)$. Necessary and sufficient conditions for self-convexity are given when $\alpha$ is $C^2$. When $\alpha(x)=d(x,\mathcal{N})^{-2}$, where $d(x,\mathcal{N})$ is the distance from $x$ to a $C^2$ submanifold $\mathcal{N}\subset\mathbb{R}^j$, we prove that $\alpha$ is self-convex when restricted to the largest open set of points $x$ where there is a unique closest point in $\mathcal{N}$ to $x$. We also show, using this more general notion, that the square of the condition number $\|A\|_F\,/\sigma_n(A)$ is self-convex in projective space and the solution variety.
['Carlos Beltrán', 'Jean-Pierre Dedieu', 'Gregorio Malajovich', 'Michael Shub']
Convexity Properties of the Condition Number