  We study the Davis-Wielandt shell and the Davis-Wielandt radius of an
operator on a normed linear space $\mathcal{X}$. We show that after a suitable
modification, the modified Davis-Wielandt radius defines a norm on
$\mathcal{L}(\mathcal{X})$ which is equivalent to the usual operator norm on
$\mathcal{L}(\mathcal{X})$. We introduce the Davis-Wielandt index of a normed
linear space and compute its value explicitly in case of some particular
polyhedral Banach spaces. We also present a general method to estimate the
Davis-Wielandt index of any polyhedral Banach space.
