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Journal of Convex Analysis 11 (2004), No. 2, 303--334
Copyright Heldermann Verlag 2004

The Generalized Minkowski Functional with Applications in Approximation Theory

Szilard Gy. Révész
A. Rényi Institute of Mathematics, Hungarian Academy of Sciences, P.O.Box 127, 1364 Budapest, Hungary,

Yannis Sarantopoulos
National Technical University, School of Applied Mathematical and Physical Sciences, Dept. of Mathematics, Zografou Campus, 15780 Athens, Greece,

We give a systematic and thorough study of geometric notions and results connected to Minkowski's measure of symmetry and the extension of the well-known Minkowski functional to arbitrary, not necessarily symmetric convex bodies K on any (real) normed space X. Although many of the notions and results we treat in this paper can be found elsewhere in the literature, they are scattered and possibly hard to find. Further, we are not aware of a systematic study of this kind and we feel that several features, connections and properties -- e.g. the connections between many equivalent formulations -- are new, more general and they are put in a better perspective now. In particular, we prove a number of fundamental properties of the extended Minkowski functional α (K, x), including convexity, global Lipschitz boundedness, linear growth and approximation of the classical Minkowski functional of the central symmetrization of the body K. Our aim is to present how in the recent years these notions proved to be surprisingly relevant and effective in problems of approximation theory.

Keywords: convex body, support function, supporting hyperplanes, halfspaces and layers, Minkowski functional, convex functions in normed spaces, Lipschitz bounds, central symmetrization, centroid, cone of convex bodies, measure of symmetry, width of K in a direction, homothetic transformations, existence of minima of continuous convex functions in normed spaces, separation of convex sets, multivariate polynomials, Bernstein and Chebyshev type extremal problems for multivariate polynomials.

MSC 2000: Primary 46B20; 41A17, 41A63, 41A44, 26D05.

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