On the duality of the symmetric strong diameter $2$ property in Lipschitz spaces

Abstract: We characterise the weak$^*$ symmetric strong diameter $2$ property in Lipschitz function spaces by a property of its predual, the Lipschitz-free space. We call this new property decomposable octahedrality and study its duality with the symmetric strong diameter $2$ property in general. For a Banach space to be decomposably octahedral it is sufficient that its dual space has the weak$^*$ symmetric strong diameter $2$ property. Whether it is also a necessary condition remains open.