Grünbaum's inequality for sections

We show∫E∩θ+f(x)dx∫Ef(x)dx≥(kγ+1(n+1)γ+1)kγ+1γ for all k-dimensional subspaces E⊂Rn, θ∈E∩Sn−1, and all γ-concave functions f:Rn→[0,∞) with γ>0, 0<∫Rnf(x)dx<∞, and ∫Rnxf(x)dx at the origin o∈Rn. Here, θ+:={x:〈x,θ〉≥0}. As a consequence of this result, we get the following generalization of Grünbaum's inequality:volk(K∩E∩θ+)volk(K∩E)≥(kn+1)k for all convex bodies K⊂Rn with centroid at the origin, k-dimensional subspaces E⊂Rn, and θ∈E∩Sn−1. The lower bounds in both of our inequalities are the best possible, and we discuss the equality conditions.