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

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.


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


Zbl 1077.51505
Full Text: DOI arXiv


[1] Cauchy, A. L., Recherches sur les polyèdres (Premier Mémoire), J. Éc. Polytech., 9 (1813)
[2] Cufí, J.; Reventós, A., Evolutes and isoperimetric deficit in two-dimensional spaces of constant curvature, Arch. Math., 50 (2014) · Zbl 1340.53004
[3] Kneser, A., Bemerkungen über die Anzahl der Extrema der Krümmung auf geschlossenen Kurven und über verwandte Fragen in einer nicht euklidischen Geometrie, (Festschrift Heinrich Weber, vol. 1 (1912), Teubner), 170-180 · JFM 43.0463.01
[4] Knill, O., A graph theoretical Poincaré-Hopf theorem (2012)
[5] Mukhopadhayaya, S., New methods in the geometry of a plane arc, Bull. Calcutta Math. Soc., 1, 31-37 (1909) · JFM 40.0624.01
[6] Musin, O., A four-vertex theorem for a polygon, Kvant, 2, 11-13 (1997)
[7] Musin, O., Curvature extrema and four vertex theorems for polygons and polyhedra, J. Math. Sci., 119 (2004) · Zbl 1077.51505
[8] Osserman, R., The four-or-more vertex theorem, Am. Math. Mon., 92, 5, 332-337 (1985) · Zbl 0579.53002
[9] Scherk, P., The four-vertex theorem, (Proc. First Canadian Math. Congress. Proc. First Canadian Math. Congress, Montreal (1945)), 97-102
[10] Singer, D., Diffeomorphisms of the circle and hyperbolic curvature, Conform. Geom. Dyn., 5, 1, 1-5 (2001) · Zbl 1039.53020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.