On special pairs of polygons with minimal area sum

Abstract

Let n be a positive integer, n ≥ 3 and C the unit circle line in ℝ². Let α_l,..., α_n be n pairwise distinct points of C, lying on C in this cyclic order, and such that the origin O is an interior point of A:= cony {a_l,..., a_n}; put α _n+1 = α₁. Finally, let B be the n-gon circumscribed to C which is obtained as the intersection of the closed supporting half-planes of C in α_l,..., α_n. When is the area sum F(A) + F(B) minimal as n and α_l,..., α_n vary? It is known that the pair of squares and only it realizes the minimum (cf. [1], [3], [5]). It is the purpose of this note to complement this result in several respects.