Convex Polygons with Parallel Opposite Sides: Convergence, Reconstruction, and Isoperimetric Inequalities

Abstract

In this paper, we study affine $\hat{I}»$-equidistants of convex polygons with parallel opposite sides (CPPOSes), introduced by M. Craizer et al. (Polygons with Parallel Opposite Sides, Discrete & Computational Geometry 50 (2013), 474-490), and relate them to affine $\hat{I}»$-equidistants of smooth convex curves. We show how, and under which conditions, the original CPPOS can be reconstructed from its Wigner caustic and centre symmetry set. We prove a discrete version of the improved isoperimetric inequality: $$L(\mathcal{P}{)}^2⩾8n\tan \left(\frac{\ddot{I}}{€}2n\right)\cdot (A(\mathcal{P})+2\left|A^{\ast }\left({\mathrm{E}}_{0.5}(\mathcal{P})\right)\right|),$$ where $\mathcal{P}$ is a CPPOS with $2n$ vertices, $L(\mathcal{P})$ is its perimeter, $A(\mathcal{P})$ is the area enclosed by $\mathcal{P}$, and $A^{\ast }\left({\mathrm{E}}_{0.5}(\mathcal{P})\right)$ denotes the oriented area of the Wigner caustic of $\mathcal{P}$. Moreover, equality holds if and only if $\mathcal{P}$ is an equiangular CPPOS of constant width. We also prove sharp area estimates for the oriented areas of the Wigner caustic and the centre symmetry set of a CPPOS. More precisely, we show that the absolute value of the oriented area of the Wigner caustic is at most one quarter of the area of $\mathcal{P}$, while the absolute value of the oriented area of the centre symmetry set is at most the area of $\mathcal{P}$. Both bounds are sharp.

34 pages, 19 figures