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 \(\ell _1^2\), 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 \(\frac{\sum _{k=1}^m k^m ~\left( {\begin{array}{c}m\\ k\end{array}}\right) }{m!}\) is the optimal upper bound for the mth polarization constant for arbitrary \(m\in \mathbb {N}\). 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.

中文翻译：

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数值半径的极化常数

我们引入并研究了Banach空间X的第m个极化常数，用于数值半径。我们首先证明了该常数与与范数相关的原始第m个极化常数之间的差异，方法是证明只有当X严格凸时，新常数才最小，并且存在一个不具有几乎等距的Banach空间。\（\ ell _1 ^ 2 \）的副本，使得数值半径的第二个极化常数最大。对于Choi和Kim问题，我们也对复杂的Banach空间给出了否定的答案（J Lond Math Soc（2）54（1）：135–147，1996）。\（\ frac {\ sum _ {k = 1} ^ mk ^ m〜\ left（{\ begin {array} {c} m \\ k \ end {array}} \ right）} {m！} \）是任意\（m \ in \ mathbb {N} \）中第m个极化常数的最佳上限。最后，我们概括了García等人的结果。（Proc Am Math Soc 142（4）：1229-1235，2014年），对于2齐次多项式，复数茂盛空间的数值指数大于或等于1/3。