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arXiv · 2610.36220

The reverse isoperimetric inequality for convex shapes included in a triangle

Rida Ait El Manssour·Zakaria Fattah·Ilias Ftouhi
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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 $η\in(0,1)$, we determine the set with maximal perimeter among convex bodies $K\subset T$ 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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Rida Ait El Manssour, Zakaria Fattah, Ilias Ftouhi. 2026-10-26. The reverse isoperimetric inequality for convex shapes included in a triangle. https://arxiv.org/abs/2610.36220

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