Viviani’s Theorem, Minkowski’s Theorem and Equiangular Polygons

Abstract

Consider a polygon P ⊂ R^2 and a positive real number t. The action of dilating (or shrinking) P by a factor of t is equivalent to dilating (or shrinking) each side of P by t, while preserving the unit normal vectors to the edges. A possible variation to this task is to consider elongating or shortening each side of P by t, also keeping the unit normal vectors intact. It is not clear a priori that such a task can always be accomplished. The current work addresses this adaptation and draws a connection with Viviani’s theorem and equiangular polygons. The main purpose of the paper is to highlight a famous theorem of Minkowski from convex geometry that makes this connection possible and gives a generalization to higher dimensions.