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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, revesz@renyi.hu
Yannis Sarantopoulos
National Technical University, School of Applied Mathematical and Physical Sciences,
Dept. of Mathematics, Zografou Campus, 15780 Athens, Greece,
ysarant@math.ntua.gr
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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