On the smallest area $n-1$-gon containing a convex $n$-gon

TL;DR

This paper establishes tight bounds for containing convex pentagons and hexagons within smaller convex polygons of minimal area, advancing understanding of polygon containment with precise area constraints.

Contribution

It proves tight area bounds for convex pentagons and hexagons contained within smaller polygons, and conjectures a general formula for larger convex polygons.

Findings

Convex pentagon contained in a quadrilateral of area ≤ 3/√5.

Convex hexagon contained in a pentagon of area ≤ 7/6.

Results are tight, as shown by regular polygons.

Abstract

We prove that every unit area convex pentagon is contained in a convex quadrilateral of area no greater than $3/\sqrt{5}$, and that every unit area convex hexagon is contained in a convex pentagon of area no greater than $7/6$. Both results are tight as the case of the regular pentagon (hexagon) shows. We conjecture that for every $n\ge 6$, every unit area convex $n$ - gon is contained in a $(n-1)$ - gon of area no greater than $1+\tan(2\pi/n)\sec(\pi/n)/n$.