有限维赋范空间中的局部Steiner问题
The local Steiner problem in finite-dimensional normed spaces
摘要：
本文给出了有限维赋范空间中某些星形构形是Steiner极小树的一般证明方法。这种方法推广了Lawlor和Morgan（1994）的结果，这些结果只能应用于可微范数。推广使用次微分从凸分析。我们将这种方法应用于两个特殊的规范。第一个规范，发生在工作的Cieslik，有单位球极的差异机构的n-单纯形（在三维这是菱形十二面体）。我们确定在这个标准的Steiner最小树中的一个给定点的最大程度。证明中主要利用了极值有限集理论。第二个范数出现在Mark Conger（1989）的工作中，是ell_1-范数与ell_2范数的小倍数之和。对于第二个范数，我们确定Steiner点的最大度。
英文摘要：
We develop a general method of proving that certain star configurations in finit e-dimensional normed spaces are Steiner minimal trees. This method generalizes the results of Lawlor and Morgan (1994) that could only be applied to differentiable norms. The generalization uses the subdifferential calculus from convex analysis. We apply this method to two special norms. The first norm, occurring in the work of Cieslik, has unit ball the polar of the difference body of the n-simplex (in dimension 3 this is the rhombic dodecahedron). We determine the maximum degree of a given point in a Steiner minimal tree in this norm. The proof makes essential use of extremal finite set theory. The second norm, occuring in the work of Mark Conger (1989), is the sum of the ell_1-norm and a small multiple of the ell_2 norm. For the second norm we determine the maximum degree of a Steiner point.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
