On a covering problem related to the centered Hardy--Littlewood maximal inequality (Q1426918)

In [\textit{A. Melas}, Trans. Am. Math. Soc. 354, No. 8, 3263--3273 (2002; Zbl 1015.42015)] and [\textit{A. Melas}, Ann. Math. (2) 157, No. 2, 647--688 (2003; Zbl 1055.42013)] it was introduced a geometric covering problem that was used to find the best possible bound in the weak type (1,1) inequality for the centered Hardy-Littlewood maximal operator. In this paper the author considers a generalization of this covering problem on the real line and finds the best possible constant associated with it. The proof is based on a discretization of the covering problem and on the establishment of an equivalent problem of combinatorial nature.