Convergence of the shadow sequence of inscribed polygons

Let P be a polygon inscribed in a circle. The shadow of P is a polygon P ′ whose vertices are at the midpoints of the arcs of consecutive points of P . The shadow sequence P , P , P , . . . is a sequence of inscribed polygons such that each P t is the shadow of P t−1 for all t ≥ 0. We show in this abstract that the shadow sequence converges to the regular polygon, and in such way that variance decreases by at least one half at every step. Our proofs extend to the more general case where instead of placing the vertices of the shadow at the ratio of 1/2 of every arc we place them at an arbitrary fixed ratio α (0 < α < 1) going in the clockwise or counterclockwise direction. Sequences of polygons generated by performing iterative processes on an initial polygon have been studied extensively in geometry. One of the most studied sequences is the one sometimes referred to as Kasner polygons. Given a polygon P , the Kasner descendent P 1 of P 0 is obtained by placing the vertices of P 1 at the midpoints of the edges of P . Fejes Tóth [6] was interested in the more general problem of sequences of Kasner polygons where each polygon P t in the sequence is obtained by dividing every edge of P t−1 with a ratio α : (1 − α) in the clockwise (or counterclockwise) direction and making the division points the vertices of P t for t = 1, 2, . . . . He proved that if α = 1/2 (Kasner polygon), then the sequence converges to a regular polygon when P 0 is a convex pentagon or a convex hexagon. He conjectured that for any α and any initial convex polygon, the sequence converges to a regular polygon. Reichardt [5] showed that if α = 1/2, every convex polygon converges to the regular polygon. Later, Lükő [4] proved that for any α ∈ [0, 1] and for any convex polygon P , the sequence P , P , P , . . . converges to a regular polygon, thus settling the more general conjecture of Fejes Tóth. More on Kasner polygons can be found in [2, 1]. The shadow sequence we study in this abstract is similar to the Kasner sequence. It is a sequence of inscribed polygons, where vertices of P t are at midpoints of the arcs between consecutive vertices of P t−1, for all t ≥ 0. Hitt and Zhang [3] show that the