LAUSR

Significance A cornerstone in the modern political organization of societies is the existence of a deliberative assembly, reflecting the needs of different population segments. As modern societies become more complex, representation according to dimensions beyond political affiliation and geography is demanded; examples include gender balance and ethnicity. As this dimensionality increases, the task becomes more challenging and requires more sophisticated mathematical tools. In this paper, we initiate the study of multidimensional apportionments and show that, in three and more dimensions, their existence is not guaranteed. However, our main result states that it is possible to elect a house nearly respecting proportionality of representation along several dimensions simultaneously. We finally illustrate the potential of our approach with recent election data.