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For p ≥ 1, Lutwak, Yang and Zhang introduced the concept of p-projection body, and Lutwak introduced the concept of Lp-affine surface area of convex body. In this paper, we develop the Minkowski-Funk transform approach in the Lp-Brunn-Minkowski theory. We consider the question of whether Π_pK ⊆ Π_pL implies Ω_p (K) ≤ Ω_p (L), where Π_pK and Ω_pK denotes the p-projection body of convex body K and the L_p-affine surface area of convex body K, respectively. We also formulate and solve a generalized L_p-Winterniz problem for Firey projections.

Lp−Winterniz problem on firey projection of convex bodies