  Let X be a closed subspace of a Banach space Y and J be the inclusion map. We
say that the pair (X,Y) has the Daugavet property if for every rank one bounded
linear operator T from X to Y the following equality \|J+T\|=1+\|T\| holds. A
new characterization of the Daugavet property in terms of weak open sets is
given. It is shown that the operators not fixing copies of l_1 on a Daugavet
pair satisfy the Daugavet equation.
  Some hereditary properties are found: if X is a Daugavet space and Y is its
subspace, then Y is also a Daugavet space provided X/Y has the Radon-Nikodym
property; if Y is reflexive, then X/Y is a Daugavet space. Becides, we prove
that if (X,Y) has the Daugavet property and Y \subset Z, then Z can be renormed
so that (X,Z) possesses the Daugavet property and the equivalent norm coincides
with the original one on Y.
