A Closed Form for the Chord-Power Integral I_2 of a Triangle

Abstract

The chord-power integrals $I_k$ 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 $I_2$ 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 $\ln 3$, attained uniquely by the equilateral triangle. The set $\{I_0,I_1,I_2\}$ identifies a triangle up to congruence, complementing J. Gates's algebraic recognition via $\{I_0,I_1,I_5\}$ with the minimal index set $\{0,1,2\}$.

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, Universität Augsburg. Section 4 (the recognition theorem for triangles from {I_0, I_1, I_2}) is not in Heinrich