I have uploaded a new manuscript to arXiv in which I prove a theorem that I stated as a conjecture in an earlier, related manuscript. The result gives a QR-like factorization for an arbitrary vector norm: the second factor is upper triangular and the condition number of the first factor, measured in the chosen norm, is bounded independently of the input matrix. For the Euclidean norm this reduces to the familiar case where the first factor is orthogonal. Other norms can bring useful properties of their own, such as the outlier resistance or sparsity promotion associated with the l1 norm.
This was a very hard theorem for me to prove, so I am especially happy that the result coincides so satisfactorily with my intuition. I give some of the thinking behind the inverse bound and the resulting factorization in a follow-up post.