A Closed Form for the Chord-Power Integral I_2 of a Triangle
Abstract
The chord-power integrals are classical integral-geometric functionals of a planar convex body, obtained by integrating powers of the chord length against the kinematic measure on the space of lines meeting the body. We establish a single-expression closed form for on an arbitrary triangle, involving logarithms symmetric in the sides, and derive two analytic consequences: a power-sum series representation, and a sharp isoperimetric-type inequality with explicit constant involving , attained uniquely by the equilateral triangle. The set identifies a triangle up to congruence, complementing J. Gates's algebraic recognition via with the minimal index set .
Keywords
Cite
@article{arxiv.2605.24616,
title = {A Closed Form for the Chord-Power Integral I_2 of a Triangle},
author = {Mher Martirosyan and Ruben Sargsyan},
journal= {arXiv preprint arXiv:2605.24616},
year = {2026}
}
Comments
Sections 2 and 3 of this paper (the closed-form expression for I_2, of a triangle, the power-sum series representation, and the sharp inequality) were obtained earlier by Lothar Heinrich, "On Lower Bounds of Second-Order Chord Power Integrals of Convex Discs," Preprint 27/2009, Universit\"at Augsburg. Section 4 (the recognition theorem for triangles from {I_0, I_1, I_2}) is not in Heinrich