Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Convex Analysis 15 (2008), No. 3, 547--560Copyright Heldermann Verlag 2008 Prox-Regularity of Spectral Functions and Spectral Sets Aris Daniilidis Dep. de Matemàtiques C1/308, Universitat Autònoma de Barcelona, 08193 Bellaterra - Cerdanyola del Vallès, Spain arisd@mat.uab.es Adrian Lewis School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY 14853, U.S.A. aslewis@orie.cornell.edu Jérôme Malick CNRS, Laboratoire J. Kunztmann, Grenoble, France jerome.malick@inria.fr Hristo Sendov Dept. of Statistical and Actuarial Sciences, The University of Western Ontario, London, Ontario, Canada hssendov@stats.uwo.ca [Abstract-pdf] Important properties such as differentiability and convexity of symmetric functions in $\mathbb{R}^{n}$ can be transferred to the corresponding spectral functions and vice-versa. Continuing to built on this line of research, we hereby prove that a spectral function $F\colon {\bf S}^n \rightarrow \mathbb{R\cup \{+\infty \}}$ is prox-regular if and only if the underlying symmetric function $f\colon\mathbb{R}^{n}\rightarrow \mathbb{R\cup \{+\infty \}}$ is prox-regular. Relevant properties of symmetric sets are also discussed. Keywords: Spectral function, prox-regular function, eigenvalue optimization, invariant function, permutation theory. MSC: 15A18, 49J52; 47A75, 90C22 [ Fulltext-pdf  (150  KB)] for subscribers only.