Authors:Mher Martirosyan (Yerevan State University and American University of Armenia), Ruben Sargsyan (American University of Armenia)
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 $\{0, 1, 2\}$.
| Comments: | 8 pages |
| Subjects: | Metric Geometry (math.MG); Classical Analysis and ODEs (math.CA); Probability (math.PR) |
| MSC classes: | 52A22, 52A40 (Primary) 53C65, 26D15, 26A48 (Secondary) |
| Cite as: | arXiv:2605.24616 [math.MG] |
| (or arXiv:2605.24616v1 [math.MG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.24616 arXiv-issued DOI via DataCite (pending registration) |
From: Mher Martirosyan [view email]
[v1]
Sat, 23 May 2026 15:00:20 UTC (10 KB)
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