  We consider a general concept of Daugavet property with respect to a norming
subspace. This concept covers both the usual Daugavet property and its weak$^*$
analogue. We introduce and study analogues for narrow operators and rich
subspaces in this general setting and apply the results to show that a quotient
of $L_1[0,1]$ over an $\ell_1$-subspace can fail the Daugavet property. The
latter answers a question posed to us by A. Pelczynski in the negative.
