PreprintCC BY 4.0arXiv
The reverse isoperimetric inequality for convex shapes included in a triangle
Abstract
We establish a sharp reverse isoperimetric inequality for convex subsets of a triangle under a prescribed area constraint. Given a triangle T and an area fraction η∈(0,1), we determine the set with maximal perimeter among convex bodies K⊂ T satisfying |K|=η|T| and characterize the equality cases. Every maximizer is a triangle, and the optimal configuration is obtained by retaining a longest side of T and placing the third vertex on a shortest side, at the position determined by the area constraint. This characterization yields an explicit expression for the maximal perimeter in terms of η and the side lengths of T.
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BibTeX
@misc{manssour2026reverse,
title = {{The reverse isoperimetric inequality for convex shapes included in a triangle}},
author = {Rida Ait El Manssour and Zakaria Fattah and Ilias Ftouhi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36220},
eprint = {2610.36220},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36220},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The reverse isoperimetric inequality for convex shapes included in a triangle
AU - Rida Ait El Manssour
AU - Zakaria Fattah
AU - Ilias Ftouhi
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.36220
UR - https://arxiv.org/abs/2610.36220
AB - We establish a sharp reverse isoperimetric inequality for convex subsets of a triangle under a prescribed area constraint. Given a triangle T and an area fraction η∈(0,1), we determine the set with maximal perimeter among convex bodies K⊂ T satisfying |K|=η|T| and characterize the equality cases. Every maximizer is a triangle, and the optimal configuration is obtained by retaining a longest side of T and placing the third vertex on a shortest side, at the position determined by the area constraint. This characterization yields an explicit expression for the maximal perimeter in terms of η and the side lengths of T.
ER -