PERIMETER ROTATIONS OF TRIANGLES AND THE ERDŐS-DEBRUNNER INEQUALITY
Abstract
Let s be the semiperimeter of a triangle ABC . Walking the same oriented distance s + ℓ along the boundary from each vertex produces points A 1 ∈ BC , B 1 ∈ C A, C 1 ∈ AB, and we prove the exact identity
where R and r are the circumradius and the inradius. It follows from a three-parameter barycentric determinant formula, recovers the equal-perimeter inequality of Aassila and the extension of Torrejón, and exhibits the invariant form of the area computation of Lupu and Spȃtaru. With Blundon's inequality it gives sharp bounds at fixed R and r . We then study the Erdős-Debrunner power-mean inequality along this family. With q = ℓ/s, the optimal exponent is found explicitly for 1/9 ≤ |q| < 1/5, it tends to the unrestricted optimum as q → 0, and as |q| ↑ 1/9 it tends to 1, with a corner at 1/9 and an explicit, super-exponentially small correction to a boundary exponent.
Related
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- A perimeter analogue of Franklin's identity and an inequality related to the parity of parts
- Rainbow Triangles and the Erdős-Hajnal Problem in Projective Geometries
- Fixed-Perimeter Franklin Statistics and an Eventual Inequality
- Noncommutative Marcus-Ree inequality and Erdős channels
BibTeX
@misc{aassila2026perimeter,
title = {{PERIMETER ROTATIONS OF TRIANGLES AND THE ERDŐS-DEBRUNNER INEQUALITY}},
author = {Mohammed Aassila},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05777110v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - PERIMETER ROTATIONS OF TRIANGLES AND THE ERDŐS-DEBRUNNER INEQUALITY
AU - Mohammed Aassila
PY - 2026
DA - 2026/10/04
PB - HAL
UR - https://hal.science/hal-05777110v1
AB - Let s be the semiperimeter of a triangle ABC . Walking the same oriented distance s + ℓ along the boundary from each vertex produces points A 1 ∈ BC , B 1 ∈ C A, C 1 ∈ AB, and we prove the exact identity where R and r are the circumradius and the inradius. It follows from a three-parameter barycentric determinant formula, recovers the equal-perimeter inequality of Aassila and the extension of Torrejón, and exhibits the invariant form of the area computation of Lupu and Spȃtaru. With Blundon's inequality it gives sharp bounds at fixed R and r . We then study the Erdős-Debrunner power-mean inequality along this family. With q = ℓ/s, the optimal exponent is found explicitly for 1/9 ≤ |q| < 1/5, it tends to the unrestricted optimum as q → 0, and as |q| ↑ 1/9 it tends to 1, with a corner at 1/9 and an explicit, super-exponentially small correction to a boundary exponent.
ER -