The Daugavet property for spaces of Lipschitz functions

For a compact metric space $K$ the space $\text{Lip}(K)$ has the Daugavet property if and only if the norm of every $f\in \text{Lip}(K)$ is attained locally. If $K$ is a subset of an $L_p$-space, $1<p<\mathrm{\infty }$, this is equivalent to the convexity of $K$.