The finite-dimensional problem of embedding a given compact D ⊂ R p into the lower Lebesgue set G(α) = {y ∈ R p : f(y) 6 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) on 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.
