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Arbitrarily weak linear convexity conditions for multivariate polynomials. (English) Zbl 0974.41022

Summary: The article presents a general construction of linear sufficient convexity conditions for multivariate polynomials in Bernstein-Bézier representation over simplices. As the main new feature of the construction, the obtained conditions can be made as weak as desired; they can be adapted to any finite set of strongly convex polyomials. Other linear convexity conditions, e.g. those derived by G. Chang and P. J. Davis [J. Approximation Theory 40, 11-28 (1984; Zbl 0528.41005)], or by J. M. Carnicer, M. S. Floater, and J. M. Peña [Comput. Aided Geom. Des. 15, No. 1, 27-38 (1997; Zbl 0894.68151)], appear as special cases of the general construction. The article includes a quantitative comparison of the various linear convexity criteria.

MSC:

41A63 Multidimensional problems
41A10 Approximation by polynomials
26B25 Convexity of real functions of several variables, generalizations
65D17 Computer-aided design (modeling of curves and surfaces)
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