We study the Davis-Wielandt shell and the Davis-Wielandt radius of an operator on a normed linear space X. We show that after a suitable modification, the modified Davis-Wielandt radius defines a norm on L(X) which is equivalent to the usual operator norm on L(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 finite-dimensional Banach space.