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In this paper, we introduce the concept of general $L_p$ projection body andgeneral $L_p$ centroid body of general measures with positive homogeneitydensity function, and prove the corresponding extreme inequalities. Meanwhile,we also study their measure comparison problem and monotone inequalities.

Extreme inequalities of general $L_p$ $\mu$-projection body and general $L_p$ $\mu$-centroid body