A generalization of the Discrete Isoperimetric Inequality for Piecewise Smooth Curves of Constant Geodesic Curvature
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GENERALIZATION OF THE DISCRETE ISOPERIMETRIC INEQUALITY FOR PIECEWISE SMOOTH CURVES OF CONSTANT GEODESIC CURVATURE ´zs Csiko ´ s1 (Budapest), Zsolt La ´ngi 2 (Calgary) and Bala 3 ´rton Naszo ´ di (Calgary) Ma Dedicated to K´ aroly Bezdek on his 50th birthday 1
Dept. of Geometry, E¨ otv¨ os University, P´ azm´ any P´eter S´et´ any 1/c, 1117 Budapest, Hungary, E-mail: [email protected]
2
Dept. of Math. and Stats., University of Calgary, 2500 University Drive NW, Calgary, Ab, T2N 1N4 Canada, E-mail: [email protected]
3
Dept. of Math. and Stats., University of Calgary, 2500 University Drive NW, Calgary, Ab, T2N 1N4 Canada, E-mail: [email protected] (Received: March 17, 2005; Accepted: November 21, 2005) Abstract The discrete isoperimetric problem is to determine the maximal area polygon with at most k vertices and of a given perimeter. It is a classical fact that the unique optimal polygon on the Euclidean plane is the regular one. The same statement for the hyperbolic plane was proved by K´ aroly Bezdek [1] and on the sphere by L´ aszl´ o Fejes T´ oth [3]. In the present paper we extend the discrete isoperimetric inequality for “polygons” on the three planes of constant curvature bounded by arcs of a given constant geodesic curvature.
1. Introduction Throughout this paper, M denotes any of the three geometries of constant sectional curvature K ∈ {0, −1, 1}: the Euclidean plane, (K = 0), denoted by E2 , the hyperbolic plane, (K = −1), denoted by H2 , or the sphere, (K = 1), denoted by S2 . If a and b are two points of M, which are not antipodal if M = S2 , then ab denotes the shortest geodesic segment connecting them. The discrete isoperimetric problem is to determine the maximal area polygon with at most k vertices and of a given perimeter. It is a classical fact that the unique Mathematics subject classification number: 51M10, 51M16, 51M25. Key words and phrases: isoperimetric inequality, Euclidean plane, hyperbolic plane, spherical geometry. The authors were partially supported by the Hung. Nat. Sci. Found. (OTKA), grant no. T047102, T043556 and T037752. 0031-5303/2006/$20.00 c Akad´ emiai Kiad´ o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
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´ s, z. la ´ ngi and m. naszo ´ di b. csiko
optimal polygon for M = E2 is the regular one. The same statement for M = H2 was proved by K´ aroly Bezdek [1] and for M = S2 by L´ aszl´o Fejes T´oth [3]. We refer to these results as the (Euclidean, hyperbolic or spherical) discrete isoperimetric inequality. On discrete isoperimetric problems see also [4], [6] and [7]. The following question was asked by K´ aroly Bezdek in personal communication. Can one extend these results to circle-polygons i.e. “polygons” bounded by circular arcs of a given radius (instead of line segments)? Convex circle-polygons arise naturally as intersections of finitely many disks of the same radius. This paper answers the question for a slightly more general class of planar figures defined as follows. Definition. Let Γ ⊂ M be a simple closed polygon in M and let kg ≥ 0
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