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 $f \in Lip(K)$ is attained locally. If $K$ is a subset of an $L_p$-space, $1<p<∞$, this is equivalent to the convexity of $K$.