Clouds of planar convex bodies
A deep cloud is a family of convex bodies \(K_1,\dots, K_n\) arranged around a given convex body \(K\) such that the bodies \(K_1,\dots, K_n\) are pairwise non-overlapping and each ray, emanating from \(K\), intersects some of the \(K_i\) at its interior. The authors are concerned with the minimum number \(C(K)\) of congruent copies of a given convex body \(K\) in the plane yielding a deep cloud around \(K\). After shortly confirming \(C(K)\geq 4\) (with various cases of equality), they prove that \(C(K)\leq 9\), with equality iff \(K\) is a circle. Furthermore, they show that \(C(K)= 8\) if there are two concentric circles \(C_1\), \(C_2\) of radius \(r_1= 0,99\) and \(r_2= 1\), respectively, such that \(C_1\subseteq K\subseteq C_2\).
scientific article; zbMATH DE number 1761060