Melzak's Conjecture seeks the polyhedron with minimal perimeter for a given volume. In studying this problem, the first approach may be to compare different polyhedra with a unit volume. However, the wide array of volume formulas for polyhedra make a unit volume computationally cumbersome. Instead, a more efficient approach is to consider the ratio of edge length to volume. In this manner, one may assign the edge length or volume to conveniently fit the situation. This paper summarizes previous work on the problem and presents some experimental observations from recent research. Much work is presented on ideas centered around prisms and their properties, and ideas for future consideration are explained, offering contributions to the ongoing study of Melzak's Conjecture.

Author Bio

Kaitlyn and Adam graduated from Lee University in 2017. They are graduate students at the University of Alabama at Birmingham, where they are both working toward their Ph.D. in Applied Mathematics.

Austin graduated with honors from Lee University in May 2017. He is a Graduate Teaching Assistant for Austin Peay State University, where he is working toward a master’s in Mathematical Finance. He hopes to work in the finance industry or in actuarial science. Outside mathematics, he enjoys playing piano and practicing martial arts.