The Four-or-More Vertex Theorem
The Four-or-More Vertex Theorem
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四或更多顶点定理
DOI：
10.1080/00029890.1985.11971614
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发表时间：
1985
影响因子：
0.5
通讯作者：
R. Osserman
中科院分区：
文献类型：
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作者：
R. Osserman
四顶点定理指出，平面上的平滑乔丹曲线至少有四个顶点。顶点是曲率的局部最大值或最小值。因此，椭圆恰好有四个顶点，分别位于长轴和短轴的末端。在微分几何导论（[2]、[5]、[6]、[7]、[13]、[16]、[21]）中，在曲线是凸的附加假设下，该定理经常被证明，作为需要全局参数而不是纯粹局部参数的定理的早期实例。四顶点定理（Vierscheitelsatz，Theoreime des quatre sommets）有着悠久的历史，始于 1909 年 Mukhopadhaya [18]，他针对凸曲线陈述并证明了该定理。随后出现了一系列不同的证明、概括和类比（参见参考文献中的示例），其中包括 Gluck [9] 最近做出的一项有趣的贡献，他证明了一种逆命题。因此，令人有些惊讶的是，这里提出的论点似乎不仅是新的，而且比通常的证明有许多优点：
The four-vertex theorem states that a smooth Jordan curve in the plane has at least four vertices. A vertex is a local maximum or minimum of the curvature. Thus, an ellipse has exactly four vertices, at the ends of the major and minor axes. This theorem is frequently proved, under the additional assumption that the curve is convex, in introductory differential geometry ([2], [5], [6], [7], [13], [16], [21]) as an early instance of a theorem requiring global rather than purely local arguments. The four-vertex theorem (Vierscheitelsatz, Theoreime des quatre sommets) has a long history, starting in 1909 with Mukhopadhaya [18], who stated and proved it for convex curves. There followed a succession of different proofs, generalizations, and analogies (see the References for a sample), including an interesting recent contribution due to Gluck [9], who proved a kind of converse. It is therefore somewhat surprising that the argument presented here seems not only to be new, but also to have a number of advantages over the usual proofs: