A bounded linear operator T on a locally convex space X is balanced convex-cyclic if there exists a vector x 2 X such that the balanced convex hull of orb(T; x) is dense in X.A balanced convex-polynomial is a balanced convex combination of monomials f1; z; z²; z³; : : : g.In this paper we prove that the balanced convex-polynomials are dense in Lp() when ([-1; 1]) = 0.Our results are used to characterize which multiplication operators on various real Banach spaces are balanced convex-cyclic.Also,it is shown for certain multiplication
operators that every nonempty closed invariant balanced convex-set is a closed invariant subspace.

Keywords: Balanced convex-polynomials, Balanced convex-cyclic operators, Hahn Banach theorem

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