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The four-vertex theorem of a plane curve and its generalizations. (Russian. English summary) Zbl 1031.53500

The author discusses both the classical four-vertex theorem and its recently discovered space version which says that every smooth closed curve in the Euclidean 3-space which is weakly convex and has nonzero curvature has at least 4 flattening points. The latter theorem was first published in [V. D. Sedykh, A theorem on four vertices of a convex space curve. Funct. Anal. Appl. 26, No. 1, 28-32 (1992; Zbl 0777.53004)].
The article is based on a lecture given by the author for secondary school teachers in the framework of the “International Soros Science Educational Program” and is aimed at presenting contemporary developments in mathematics on an elementary level understandable for secondary school teachers and pupils.

MSC:

53A04 Curves in Euclidean and related spaces
00A06 Mathematics for nonmathematicians (engineering, social sciences, etc.)
51-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to geometry

Citations:

Zbl 0777.53004
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