Firey-Shephard problems for homogeneous measures

We introduce a concept of Firey-projection function for μ-measure, which is an extension of Lp-projection body. We consider the Firey-Shephard problem for measures, and connect it with Lp-projection body. Using this connection, we give a negative answer to the Firey-Shephard problem for homogeneous measures for all p>1, p≠n and n≥2. Moreover, we also prove a measure comparison theorem.