The relationship between the Chebyshev center of a closed convex bounded subset and a center of an inscribed ball for some “dual” set (a ball of maximal radius which is contained in the set) is considered in a real Hilbert space. The inscribed ball is unique in the discussed situation. We present an approximate algorithm for calculation of the Chebyshev center (or the center of the inscribed ball for the “dual” set) for a convex compact subset from Rn which is given via its supporting function. We reduce the problem to the solution of a linear programming problem and estimate the error between an approximate and the exact solutions in terms of the step of a grid. Few numerical examples are considered. The considered algorithm is efficient in the space of small dimension n.