Suppose \(\mathbb{K}\) is either \(\mathbb{C}\) or \(\mathbb{R}\). The non-increasing rearrangements is a \(\lVert \cdot\rVert_\infty\) contraction, i.e. for \(x,y\in \mathbb{K}^N\) we have

\[ \lVert x^*-y^*\rVert_\infty \le \lVert x-y\rVert_\infty. \]

[1, Lemma 1.14]

References Link to heading

1. I. Veselic. Class Lecture, Topic: Compressive Sensing. Technische Universität Dortmund, 2017.