We propose a new class of multidimensional poverty indices. Aggregation of the different dimensions relies on individual preferences. The Pareto principle is, therefore, satisfied among the poor. The indices add up individual measures of poverty that are computed as a convex transform of the fraction of the poverty line vector to which the agent is indifferent. The axiomatic characterization of this class retains the usual axioms of focus and subgroup consistency and it introduces additional principles for interpersonal poverty comparisons and for inequality aversion among the poor. We illustrate our approach with Russian survey data between 1995 and 2005. We find that taking preferences into account leads to considerable differences in the identification of the poor compared to standard poverty measures.