Basic version of the question: Consider a unit area planar convex region $C$ and its maximal area inscribed triangle $T(C)$. Which shape of unit area $C$ minimizes the area of its $T(C)$?

General version: Replace triangle by $n$-gon - for each value of $n$, we can ask for the shape of the unit area $C$ that has the smallest max-area inscribed $n$-gon.

An inside out version: Given a convex $n$-gon, to find the largest convex region such that the given $n$-gon is the largest inscribed $n$-gon inside it.

Analogous questions can be asked with perimeter replacing area.

Basic version of the question: Consider a unit area planar convex region $C$ and its maximal area inscribed triangle $T(C)$. Which shape of unit area $C$ minimizes the area of its $T(C)$?

General version: Replace triangle by $n$-gon - for each value of $n$, we can ask for the shape of the unit area $C$ that has the smallest max-area inscribed $n$-gon.

An inside out version: Given a convex $n$-gon, to find the largest convex region such that the given $n$-gon is the largest inscribed $n$-gon inside it.

Analogous questions can be asked with perimeter replacing area.

Ref: https://nandacumar.blogspot.com/2022/12/isosceles-triangle-containers-inside.html?m=1