Intersecting the sides of a polygon

The uploaded paper proves exactly the equivalence the candidate addresses: for a convex n-gon (n ≥ 5), (1) persistence of convexity under S, (2) d_P = 0, and (3) G_P is affine, are equivalent (Theorem 1.1) . The paper shows 2 ⇔ 3 by a direct computation relating preservation of the line at infinity to d_P = 0 , and 1 ⇔ 3 via duality S(P)* = D(P*), a convexity–to–duality translation, and the limit-point criterion tied to Glick’s operator (with G_{P*} = G_P^*) . The model’s solution mirrors this structure and is essentially the same proof. Minor issues: it momentarily states the dual convexity condition using the interior point O rather than the line-at-infinity point L in the dual plane (the paper uses L) , and it attributes the uniqueness of the fixed point entirely to Glick, whereas the present paper supplies the uniqueness argument (Moebius-line argument) beyond the fixed-point fact from Glick . These do not alter correctness.