A Closed Form for the Chord-Power Integral I_2 of a Triangle
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910260250279936 |
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| author | Martirosyan, Mher Sargsyan, Ruben |
| author_facet | Martirosyan, Mher Sargsyan, Ruben |
| contents | 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 $\{0, 1, 2\}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_24616 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Closed Form for the Chord-Power Integral I_2 of a Triangle Martirosyan, Mher Sargsyan, Ruben Metric Geometry Classical Analysis and ODEs Probability 52A22, 52A40 (Primary) 53C65, 26D15, 26A48 (Secondary) 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 $\{0, 1, 2\}$. |
| title | A Closed Form for the Chord-Power Integral I_2 of a Triangle |
| topic | Metric Geometry Classical Analysis and ODEs Probability 52A22, 52A40 (Primary) 53C65, 26D15, 26A48 (Secondary) |
| url | https://arxiv.org/abs/2605.24616 |