Abstract: For p > 0, Lutwak, Yang and Zhang introduced the concept of L_p-polar projection body Γ_-pK of a convex body K in Rⁿ. Let p ≥ 1 and K, L \subset Rⁿbe two origin-symmetric convex bodies, we consider the question of whether Γ_-p K  \subset  Γ_-p L implies \Omega_p（L） ≤ \Omega_p（K）,where \Omega_p（K） denotes the L_p-affine surface area of K and K = Voln（K）^-1/p K. We prove a necessary and sufficient condition of an analog of the Shephard problem for the L_p-polar projection bodies. 

Key words: convex body, L_p-polar projection body, L_p-affine surface area, Fourier transform, Shephard problem