We prove versions of James' weak compactness theorem for polynomials and symmetric multilinear forms of finite type. We also show that a Banach space$X$is reflexive if and only if it admits and equivalent norm such that there exists$x$_{0}≠0 in$X$and a weak-^{*}-open subset$A$of the dual space, satisfying that$x$^{*}⊗$x$_{0}attains its numerical radius. for each$x$^{*}in$A$.

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A generalization of a result of Wermer concerning the existence of polynomial hulls without analytic discs is presented. As a consequence it is shown that there exists a Cantor set X in C3 whose polynomial hull is strictly larger than X but contains no analytic discs.

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The polygonal approximation problem is a primary problem in computer graphics, pattern recognition, CAD/CAM, etc. In $R^2$, the cone intersection method (CIM) is one of the most eÆcient algorithms for approximating polygonal curves. With CIM Eu and Toussaint, by imposing an additional constraint and changing the given error criteria, resolve the three-dimensional weighted minimum number polygonal approximation problem with the parallel-strip error criterion (PS-WMN) under $L_2$ norm. In this paper, without any additional constraint and change of the error criteria, a CIM solution to the same problem with the line segment error criterion (LS-WMN) is presented, which is more frequently encountered than the PS-WMN is. Its time complexity is $O(n^3)$, and the space complexity is $O(n^2)$. An approximation algorithm is also presented, which takes $O(n^2)$ time and $O(n)$ space. Results of some examples are given to illustrate the eÆciency of these algorithms.