Finite coverings by translates of centrally symmetric convex domains
A convex domain is a compact convex subset of the Euclidean plane with non-empty interior. Let C be a fixed convex domain which has a centre, and let q and h be the maximum area of a quadrangle and the maximum area of the hexagon, both contained in C. The area of a set K is denoted by a(K), and the area of C by a. The following theorem is proved: If a convex domain K is covered by n translates of C then \[ a(K) \leq (n-1)h + a - \frac{h-q}{2(a+q)}A, \] where \[ A = \{(a+h)^ 2+ (a+q)[4(n-1)h+3a- q]\}^{1/2}- a - h. \] This improves upon the inequality \(a(K)\leq (n- 1)h+a\) obtained by \textit{R. P. Bambah} and \textit{C. A. Rogers} [J. Lond. Math. Soc. 27, 304-314 (1952; Zbl 0046.380)]. Let \(\theta_ n(C)= \inf \{na/a(K)\), where K can be covered by n translates of \(C\}\), and let \(\theta\) (C) be the least density of covering of the plane by translates of C. The above theorem implies that \(\theta_ n(C)>\theta (C)\) for any central convex domain C other than a parallelogram, and any integer \(n\geq 26\).
scientific article; zbMATH DE number 4028294