Polarization Constant for the Numerical Radius

We introduce and investigate the mth polarization constant of a Banach space X for the numerical radius. We first show the difference between this constant and the original mth polarization constant associated with the norm by proving that the new constant is minimal if and only if X is strictly convex, and that there exists a Banach space which does not have an almost isometric copy of ℓ12, such that the second polarization constant for the numerical radius is maximal. We also give a negative answer for complex Banach spaces to the question of Choi and Kim (J Lond Math Soc (2) 54(1):135–147, 1996) whether ∑k=1mkm(mk)m! is the optimal upper bound for the mth polarization constant for arbitrary m∈ N. Finally, we generalize the result of García et al. (Proc Am Math Soc 142(4):1229–1235, 2014) that, for 2-homogeneous polynomials, the numerical index of a complex lush space is greater or equal than 1/3.