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Grant, Kyle; Mogilski, Wiktor

A discrete four vertex theorem for hyperbolic polygons. (English) Zbl 1518.51011

Discrete Math. 346, No. 7, Article ID 113371, 7 p. (2023).
The classical four-vertex theorem in differential geometry states that the curvature along a simple closed smooth curve in the plane has at least four local extrema. There are several discrete versions of the four-vertex theorem. O. Musin [“A four-vertex theorem for a polygon”, Kvant 2, 11–13 (1997)] introduced the notion of discrete curvature and proved a four-vertex theorem for Euclidean polygons. In [O. R. Musin, J. Math. Sci., New York 119, No. 2, 268–277 (2001; Zbl 1077.51505); translation from Zap. Nauchn. Semin. POMI 280, 251–271 (2001)], he introduced the notion of discrete evolute.
In the paper under review, the authors adapt the techniques of Musin and prove a discrete four-vertex theorem for convex hyperbolic polygons.
Reviewer: Elizaveta Zamorzaeva (Chişinău)

MSC:

51L15 \(n\)-vertex theorems via direct methods

Keywords:

discrete four-vertex theorem; discrete curvature; convex hyperbolic polygon; evolute

Citations:

Zbl 1077.51505
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\textit{K. Grant} and \textit{W. Mogilski}, Discrete Math. 346, No. 7, Article ID 113371, 7 p. (2023; Zbl 1518.51011)
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References:

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