Grünbaum's inequality for sections

We show [Formula presented]≥([Formula presented])^{[Formula presented]} for all k-dimensional subspaces E⊂Rⁿ, θ∈E∩Sⁿ⁻¹, and all γ-concave functions f:Rⁿ→[0,∞) with γ>0, 0<∫_Rⁿ f(x)dx<∞ and ∫_Rⁿ xf(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: [Formula presented]≥([Formula presented])^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.