Adv. Appl. Prob. (SGSA) 41, 358–366 (2009) Printed in Northern Ireland  Applied Probability Trust 2009 THE CHORD LENGTH DISTRIBUTION FUNCTION FOR REGULAR POLYGONS H. S. HARUTYUNYAN ∗ ∗∗ and V. K. OHANYAN, ∗ ∗∗∗ Yerevan State University Abstract In this paper we obtain an elementary expression for the chord length distribution function of a regular polygon. The formula is derived using δ-formalism in Pleijel identity. In the particular cases of a regular triangle, a square, a regular pentagon, and a regular hexagon, our formula coincides with the results of Sulanke (1961), Gille (1988), Aharonyan and Ohanyan (2005), and Harutyunyan (2007), respectively. Keywords: Chord length distribution function; Pleijel identity; regular polygons 2000 Mathematics Subject Classification: Primary 60D05; 52A22 1. Introduction Let G be the space of all lines g in the Euclidean plane R 2 , and let (p, ϕ), the polar coordinates of the foot of the perpendicular to g from the origin O, be the standard coordinates for a line g ∈ G. Let μ(·) be the locally finite measure on G, invariant with respect to the group of all Euclidean motions (translations and rotations). It is well known that the element of the measure up to a constant factor has the following form (see [1, p. XIII]): μ(dg) = dg = dp dϕ, where dp is the one-dimensional Lebesgue measure and dϕ is the uniform measure on the unit circle. For each bounded convex domain D, we denote the set of lines that intersect D by [D]={g ∈ G : g ∩ D  = ∅}, and we have (see [1, p. 195] and [2, p. 130]) μ([D]) =|∂ D|, where ∂ D is the boundary of D and |∂ D| stands for the length of ∂ D. A random line in [D] is one with distribution proportional to the restriction of μ to [D]. Therefore, P(A) = μ(A) |∂ D| for any Borel set A ⊂[D]. Received 1 July 2008; revision received 13 February 2009. ∗ Postal address: Department of Mathematics and Mechanics,Yerevan State University, 1 Alex Manoogian Street, Yerevan 0025, Armenia. ∗∗ Email address: hrach87@web.am ∗∗∗ Email address: victo@aua.am 358 https://www.cambridge.org/core/terms. https://doi.org/10.1239/aap/1246886615 Downloaded from https://www.cambridge.org/core. IP address: 3.235.21.12, on 21 May 2020 at 09:17:27, subject to the Cambridge Core terms of use, available at