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

Abstract

In this paper, we study affine $λ$-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 $λ$-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: $$L(\mathcal{P})^2\geqslant 8n\tan\left(\fracπ{2n}\right)\cdot\big( A(\mathcal P)+2\left|A^\ast\left(\mathrm{E}_{0.5}(\mathcal P)\right)\right|\big),$$ 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.