The finite-dimensional problem of embedding a given compact D ⊂ R ^p into the lower Lebesgue set G(α) = {y ∈ R ^p: f(y) <= α} of the convex function f(·) with the smallest value of α due to the offset of D is considered. Its mathematical formalization leads to the problem of minimizing the function φ(x) = max _y∈D f(y − x) на R ^p. The properties of the function φ(x) are researched, necessary and sufficient conditions and conditions for the uniqueness of the problem solution are obtained. As an important case for applications, the case when f(·) is the Minkowski gauge function of some convex body M is singled out. It is shown that if M is a polyhedron, then the problem reduces to a linear programming problem. The approach to get an approximate solution is proposed in which, having known the approximation of x_i to obtain x_i+1 it is necessary to solve the simpler problem of embedding the compact set D into the Lebesgue set of the gauge function of the set M_i = G(α_i), where α_i = f(x_i). The rationale for the convergence for a sequence of approximations to the problem solution is given.