Abstract

The impetus for our work was a preprint by Alathea Jensen, titled self-polar polytopes. In the preprint, Jensen describes an intriguing method to add vertices to a self-polar polytope while maintaining self-polarity. This method, applied exclusively to self-nolar planar polytopes, is our main focus for our work here. We expound upon the method, as well as clarify the underlining theoretical framework it was derived from. In doing so, we have built up our own set-up and framework and proved the theoretical steps independently, often differently than the original paper. In addition, we prove some noteworthy properties of self-nolar sets such as: all self-nolar sets are convex, the family of all self-nolar sets is uncountable, and the set of all self-nolar planar polytopes is dense in the set of all self-nolar planar sets. We also give proofs concerning the length of the boundary of a self-nolar set with smooth boundary, the center of mass of self-nolar polytopes and the Mahler volume product. Moreover, we prove an original theorem that can be used as a practical method to construct self-nolar polytopes.