Centrality indices are ways to measure the importance of nodes in a graph; this need is so obviously relevant that it was discussed many times in sociology, psychology, mathematics and computer science, giving rise to a whole zoo of definitions of centrality. The ideas underlying such definitions are wildly different, but many centrality measures are based on shortest-path distances: such centralities are referred to as geometric. Albeit geometric centralities can use the shortest-path–length information in many different ways, most of the existing geometric centralities can be defined as a linear transformation. In this paper we define formally the class of linear geometric centralities in its full generality, and study its main properties and expressivity.