Towards multi-dimensional localisation

Localisation is a powerful tool in proving and analysing various geometric inequalities, including isoperimertic inequality in the context of metric measure spaces. Its multi-dimensional generalisation is linked to optimal transport of vector measures and vector-valued Lipschitz maps. I shall present recent developments in this area: a partial affirmative answer to a conjecture of Klartag concerning partitions associated to Lipschitz maps on Euclidean space, and a negative answer to another conjecture of his concerning mass-balance condition for absolutely continuous vector measures. During the course of the talk I shall also discuss an intriguing notion of ghost subspaces related to the above mentioned partitions.