
Journal of Convex Analysis 17 (2010), No. 2, 485507 Copyright Heldermann Verlag 2010 On Malamud Majorization and the Extreme Points of its Level Sets Pal Fischer Dept. of Mathematics and Statistics, University of Guelph, Guelph, Ontario N1G 2W1, Canada pfischer@uoguelph.ca Hristo Sendov Dept. of Statistical and Actuarial Sciences, University of Western Ontario, London, Ontario N6A 5B7, Canada hssendov@stats.uwo.ca [Abstractpdf] \baselineskip=13pt \newcommand{\R}{\mathbb{R}} We consider two types of majorization relationships between sequences of vectors $y=(y_k)_{k=1}^m$ and $x=(x_k)_{k=1}^\ell$ in $\R^n$ with $\ell\le m$. It is said that $x$ is majorized by $y$, $x \prec y$, if the sum of any $k$ vectors from $x$ is in the convex hull of all possible sums of $k$ vectors from $y$. It is said that $x$ is doubly stochastically majorized by $y$, $x \prec_{\rm ds} y$, if $x_k = \sum_{j=1}^m m_{kj}y_j$, $k=1,...,\ell$, for some doubly stochastic matrix $M=(m_{kj})_{k,j=1}^{m,m}$. \par In a recent article ["Inverse spectral problem for normal matrices and the GaussLucas Theorem", Trans. Amer. Math. Soc. 357(10) (2004) 40434064] S. M. Malamud formulated the problem of finding a geometric condition guaranteeing that $x\prec y \Leftrightarrow x \prec_{\rm ds} y$. We answer this question in the case when the vectors in $y$ are distinct and are extreme points of their convex hull. In particular, we derive a geometric characterization of the extreme points of the level set $L^2_{\prec}(y)=\{x : x \prec y\}$. Finally, we derive a set of algebraic conditions that characterize the extreme points of $L^\ell_{\prec}(y)=\{x : x \prec y\}$ for any $\ell \le m$ and $y$. Keywords: Convex set, doubly stochastic majorization, Malamud majorization, extreme point, convex function, CVS class. MSC: 52B11, 42A20 [ Fulltextpdf (217 KB)] for subscribers only. 