Given a hyperspectral image of a scene, unsupervised linear hyperspectral unmixing (HU) is a blind source separation problem aimed at identifying the spectral signatures of the materials present in the scene, termed endmembers, and, for each pixel, the abundances of each material. HU is often very challenging, namely owing to the high correlation between the spectral signatures that is commonly observed in real hyperspectral scenes. However, such ill-conditioning is often ignored in the design of existing HU methods. To the best of our knowledge, existing preconditioning techniques rely on the pure-pixel assumption, which is, however, often violated in practical scenarios. In this paper, we propose a new theoretical framework wherein a provable preconditioning method is developed without requiring the presence of pure pixels. Our approach is to identify John’s maximum-volume data-inscribed ellipsoid, followed by a specific affine transformation that maps the John’s ellipsoid into an Euclidean ball. Provably, such transformation converts the original hyperspectral vectors into a preconditioned dataset whose endmember matrix is semi-unitary, which is exploited as regularization information. Furthermore, this transformation is robust to noise and largely simplifies the ensuing unmixing/optimization procedure. Our numerical experiments provide evidence of the superior unmixing performance of the proposed method.