The Daugavet property for spaces of Lipschitz functions - Ivakhno, Yevgen; Kadets, Vladimir; Werner, Dirk (Mathematica Scandinavica, 2007)
For a compact metric space K the space Lip(K) has the Daugavet property if and only if the norm of every f Lip(K) is attained locally. If K is a subset of an L_p-space, 1<p<∞, this is equivalent to the convexity of K.