Arbitrarily weak linear convexity conditions for multivariate polynomials

Studia Scientiarum Mathematicarum Hungarica

Volumn 36, Issue 1-2, 2000, Pages 165-183

  Juttler B ^a  

^a DARMSTADT UNIVERSITY OF TECHNOLOGY   (Germany)

Author keywords

Bernstein B zier representation; Convexity; Multivariate polynomials

Indexed keywords

EID: 0042225158     PISSN: 00816906     EISSN: None     Source Type: Journal    
DOI: 10.1556/sscmath.36.2000.1-2.13     Document Type: Article

References (18)

1. CARNICER, J. M., FLOATER, M. S. and PEÑA, J. M., Linear convexity conditions for rectangular and triangular Bernstein-Bézier surfaces, Comput. Aided Geom. Design 15 (1997), 27-38. MR 98m:65023
2. CHANG, G. Z. and DAVIS, P. J., The convexity of Bernstein polynomials over triangles, J. Approx. Theory 40 (1984), 11-28. MR 85c:41001
3. CHANG, G. Z. and FENG, Y. Y., An improved condition for the convexity of Bernstein-Bézier surfaces over triangles, Comput. Aided Geom. Design 1 (1984), 279-283. Zbl 563. 41009
4. DAHMEN, W., Convexity and Bernstein-Bézier polynomials, Curves and surfaces (Chamonix Mont Blanc, 1990), P. J. Laurent, A. Le Méhauté and L. L. Schumaker, eds., Academic Press, Boston, MA, 1991, 107-134, MR 92h:65018
5. DAHMEN, W. and MICCHELLI, C. A., Convexity of multivariate Bernstein polynomials and box spline surfaces, Studia Sci. Math. Hungar. 23 (1988), 265-287. MR 90g:41005
6. FENG, Y. Y, CHEN, F. L. and ZHOU, H. L., The invariance of weak convexity conditions of B-nets with respect to subdivision, Comput. Aided Geom. Design 11 (1994), 97-107. MR 94k:65023
7. FLETCHER, R., Practical methods of optimization, Second edition, John Wiley & Sons, Ltd., Chichester, 1987. MR 89j:65050
8. GOODMAN, T. N. T., Shape preserving representations, Mathematical methods in computer aided geometric design (Oslo, 1988), T. Lyche and L. L. Schumaker, eds., Academic Press, Boston, MA, 1989, 333-351. MR 91a:65031
9. GOODMAN, T. N. T., Convexity of Bézier nets on triangulations, Comput. Aided Geom. Design 8 (1991), 175-180. MR 92a:65064
10. GRANDINE, T. A., On convexity of piecewise polynomial functions on triangulations, Comput. Aided Geom. Design 6 (1989), 181-187. MR 90k:65040
11. GREGORY, J. A. and ZHOU, J. W., Convexity of Bézier nets on sub-triangles, Comput. Aided Geom. Design 8 (1991), 207-211. MR 92g:65023
12. HE, T.-X., Shape criteria of Bernstein-Bézier polynomials over simplexes, Comput. Math. Appl. 30 (1995), 317-333. MR 96j:65012
13. LAI, M.-J., Some sufficient conditions for convexity of multivariate Bernstein-Bézier polynomials and box spline surfaces, Studia Sci. Math. Hungar. 28 (1993), 363-374. MR 95d: 65018
14. MICCHELLI, C. A. and PINKUS, A., Some remarks on nonnegative polynomials on polyhedra, Probability, statistics and mathematics, T. Anderson, K. Athreya and D. Iglebert, eds., Academic Press, Boston, MA, 1989, 163-186. MR 91h:26014
15. PRAUTZSCH, H., On convex Bézier triangles, RAIRO Model. Math. Anal. Numer. 26 (1992), 23-36. MR 93a:65019
16. SEIDEL, H.-P., Symmetric recursive algorithms for surfaces: B-patches and the de Boor algorithm for polynomials over triangles, Constr. Approx. 7 (1991), 257-279. MR 92c:41010
17. WANG, Z. B. and LIU, Q. M., An improved condition for the convexity and positivity of Bernstein-Bézier surfaces over triangles, Comput. Aided Geom. Design 5 (1988), 269-275. MR 90b:41016
18. WILLEMANS, K. and DIERCKX, P., Surface fitting using convex Powell-Sabin splines, J. Comput. Appl. Math. 56 (1994), 263-282. MR 96d:65040