Lower bounds for norms of products of polynomials

Let P-1,...,P-n be polynomials in one or several real or complex variables. Several authors, working with a variety of norms, have given estimates for a constant ill depending only on the degrees of P-1,...,P-n such that parallel to P(1)parallel to...parallel to P(n)parallel to less than or equa l to M parallel to P-1...P(n)parallel to. In this paper we show that inequalities of this type are valid for polynomi als on any complex Banach space. Our method provides optimal constants. We also derive analogous inequalities for polynomials on real Banach spaces , but the constants we obtain are generally not optimal. The search for opt imal constants does however lead to an interesting open problem in Hilbert space geometry. When we restrict attention to products of linear functionals, we find new c haracterizations of complex l(1)(n) (n greater than or equal to 2), real l( 1)(2), and real Hilbert space.