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Journal of Convex Analysis 17 (2010), No. 2, 485--507
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

Hristo Sendov
Dept. of Statistical and Actuarial Sciences, University of Western Ontario, London, Ontario N6A 5B7, Canada


\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 Gauss-Lucas Theorem", Trans. Amer. Math. Soc. 357(10) (2004) 4043--4064] 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

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