On condition numbers and the distance to the nearest ill-posed problem

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Perturbation theory for homogeneous polynomial eigenvalue problems (2003, background): “All these results are put in a more general setting by Dedieu in [4] and by Demmel in [7] using differential inequalities.”
Tensor Rank and the Ill-Posedness of the Best Low-Rank Approximation Problem (2006, unknown): “This process of forcing a solution to an ill-posed problem is almost always guaranteed to be ill-conditioned because of the infamous rule of thumb in numerical analysis [22, 23, 24]: A well-posed problem near to an ill-posed one is ill-conditioned.”
Randomized Methods for Linear Constraints: Convergence Rates and Conditioning (2008, unknown): “We are typically concerned with relative condition numbers as introduced by Demmel [10].”
Accurate SVDs of polynomial Vandermonde matrices involving orthonormal polynomials (2006, background): “Another way to look at this is that the rel_gapx is the (relative) distance from VP (x) to the nearest singular matrix, or ill-posed problem [2], since the matrix is singular if and only if two xi are equal.”