On plane coverings with convex domains

Abstract

The following theorem has been proved by Bambah, Rogers and Zassenhaus [1], Theorem A. Let K be a closed convex domain with a centre. Let A₀ A₁,..., A_n = A₀, A_n+1,....., A_n+m, be points such that: (i) the polygon A₀ A₁ ... A_n is a Jordan polygon bounding a closed domain π of area a(π ); (ii) for each r, 0 ≤ r ≤ n, there is a point common to K + A_n-1 and K + A_r; (iii) the points A_n+1, .... A_n+m are in the interior of π (iv) for each point X of π, there exists an A_r, 1 ≤ r ≤ n + m, such that X ∈ K + A_r and the line segment XA_r is in π. Then α(π) ≤ (2m+n-2)t(k), where t(K) is the area of the largest triangle contained in K.