Affine reflection transforms of convex polygons with parallel opposite sides

Abstract

We study affine reflection transforms of convex polygons with parallel opposite sides. An opposite-side involution sends a pencil through an interior point to a piecewise smooth family of lines. Its completed envelope consists of conic arcs and transition segments. The completed transform has non-positive oriented area. Maximising the negative oriented area and dividing by the polygonal area gives an affine invariant that vanishes precisely for centrally symmetric polygons. Combining an identity for reflected chord vectors with the Rogers--Shephard inequality shows that this invariant is strictly less than $1$, with optimal universal constant $1$. The energy extends continuously to the boundary and vanishes there. We also compare chord counts under compatible smoothing and prove Hausdorff convergence to the smooth transform.

arXiv comment24 pages, 5 figures; Version 1 has been split into two papers. This version (v2) treats the polygonal case