On the smallest (n-1)-gon containing a convex n-gon

Grace Park

Overview

We study the following geometric optimization problem: given a convex disk K of area 1 and a positive integer n greater than or equal to 3, what can be said about the n-sided convex polygon of a minimum area that contains K in its interior? It is known that every unit area convex disk is contained in a triangle of an area at most 2 and this result is optimal as a parallelogram requires an enclosing triangle of at least twice its area. For polygonal enclosures with four or more sides, no exact results are known. For n=4, Chakerian proved that every unit area convex disk is contained in a quadrilateral of an area no greater than sqrt(2). On the other hand, Kuperbergnoted that every quadrilateral containing a unit area pentagon must have an area of at least 3/sqrt(5) and conjectured that this is the worst-case scenario. We prove that every unit area convex pentagon is contained in a convex quadrilateral of an area no greater than 3/sqrt(5), thus providing a partial confirmation of Kuperberg's conjecture. We also show that every unit area convex hexagon is contained in a convex pentagon of an area no greater than 7/6. Both results are tight as the case of the regular pentagon and hexagon shows.

Video

From the student

Last year, I read a news article about the NASA 2020 Perseverance Rover which, in fact, landed on Mars. After reading this article, I started to think about the mechanics behind this robot. Along their journey, to navigate around numerous obstacles, the robots have to identify their shapes and find the most efficient way. Therefore, I started to ponder how to create the minimum-area barrier around a convex object. My inquisition ultimately led to my mathematical project under the guidance of Professor Ismailescu. We worked on the topic “On the smallest (n-1)-gon containing a convex n-gon.”

Completing the project, I decided to present my findings at the New Jersey Academy of Science Research Competition, and I luckily placed first in the computer science/mathematics category. Thus, I was given the honorable opportunity to participate and share my project at the 2022 American Junior Academy of Science Annual Conference.

After that project, I was attracted to the incomplete area of mathematics. I realized that by delving into true understanding, I do have the capability of contributing to the field of mathematics. Overall, this project was life-changing for me because I was able to stretch my abilities and immerse myself in the beauty of mathematics.

From the student

The animation shows that the quadrilateral enclosure is the most efficient when there is a line parallel to the base of the pentagon, and the two sides are extended to meet that parallel line (red color).