This paper addresses two problems, the inverse 1-center problem on the line with closed-interval facilities and the inverse 1-center problem on Rd. For the first problem, we develop a combinatorial O(nlogn) algorithm based on the convexity of the objective function, where n is the number of facilities. We also discuss the corresponding problem on interval graphs with the similar solution approach. Concerning the inverse 1-center problem on Rd, we propose an O(dn2logn) algorithm based on the optimality criterion, where n is the number of existing points.