The local diameter two property and the diameter two property in spaces of Lipschitz functions

We separate the local diameter two property from the diameter two property and their weak-star counterparts from each other in spaces of Lipschitz functions. We characterise the $w^{\ast }$-LD$2$P, the $w^{\ast }$-D$2$P, the LD$2$P, and the SD$2$P in these spaces. We introduce a generalised version of cyclical monotonicity to study functionals on spaces of Lipschitz functions.