|The adjusted headcount ratio M0 of Alkire and Foster (2011a) is increasingly being adopted by countries and international organizations to measure poverty. Three properties are largely responsible for its growing use: Subgroup Decomposability, by which an assessment of subgroup contributions to overall poverty can be made, facilitating regional analysis and targeting; Dimensional Breakdown, by which an assessment of dimensional contributions to overall poverty can be made after the poor have been identified, facilitating coordination; and Ordinality, which ensures that the method can be used in cases where variables only have ordinal meaning. Following Sen (1976), a natural question to ask is whether sensitivity to inequality among the poor can be incorporated into this multidimensional framework. We propose a Dimensional Transfer axiom that applies to multidimensional poverty measures and specifies conditions under which poverty must fall as inequality among the poor decreases. An intuitive transformation is defined to obtain multidimensional measures with desired properties from unidimensional FGT measures having analogous properties; in particular, Dimensional Transfer follows from the standard Transfer axiom for unidimensional measures. A version of the unidimensional measures yields the M-gamma class Mγ/0 containing the multidimensional headcount ratio for γ=0, the adjusted headcount ratio M0 for γ=1, and a squared count measure for γ=2, satisfying Dimensional Transfer. Other examples show the ease with which measures can be constructed that satisfy Subgroup Decomposability, Ordinality, and Dimensional Transfer. However, none of these examples satisfies Dimensional Breakdown. A general impossibility theorem explains why this is so: Dimensional Breakdown is effectively inconsistent with Dimensional Transfer. Given the importance of Dimensional Breakdown for policy analysis, we suggest maintaining the adjusted headcount ratio as a central measure, augmented by the squared count measure or other indices that capture inequality among the poor. The methods are illustrated with an example from Cameroon.