Convexifying of polynomials by convex factor - Abdullah, Abdulljabar Naji; Rosiak, Klaudia (Analitic and Algebraic Geometry 4, 2022)
Let X ⊂ Rn be a convex closed and semialgebraic set and b : Rn → (0,+∞) be a C 2 class positive strongly convex function. Let f be a polynomial positive on X. If X is compact, we prove that there exists an exponent N ≥ 1, such that for any ξ ∈ X, the function φN,ξ(x) = bN(x−ξ)f(x) is strongly convex on X. If X = ξ ∈ Rn : f(ξ) ≤ r is bounded we define a mapping κN : X ∋ ξ 7→ argminX φN,ξ ∈ Rn, where argminX φN,ξ is the unique point x ∈ X at which φN,ξ has a global minimum. We prove that κN is a mapping of class C 1 of X onto Y = κN(X) ⊂ X and that for any ξ ∈ X the limit of the iterations limν→∞ κν P N(ξ) exists and belongs to the set f of critical points of f. If additionally b is logarithmically strongly convex then κN is injective and it is defined on Rn, provided f takes only positive values and the leading form of f is positive except of the origin. In the case b(x) = exp |x|2 and f|X has only one critical value we prove that the mapping X ∋ ξ 7→ limν→∞ κν N(ξ) ∈ Σf ∩ X is continuous. Moreover, assuming that limν→∞ κν N(ξ) = 0 we study convergence of the sequence of the spherical parts of κν N(ξ), ν ∈ N.