Grünbaum's inequality for projections

Let K be a convex body in Rⁿ whose centroid is at the origin, let E∈G(n,k) be a subspace, and let ξ∈Sⁿ⁻¹. We find the best constant c=(kn+1)^k so that |(K|E)∩ξ⁺|_k≥c|K|E|_k, and completely determine the minimizer. Here, |⋅|_k is k-dimensional volume, K|E is the projection of K onto E, and ξ⁺={x∈Rⁿ:〈x,ξ〉≥0}. Our result generalizes both Grünbaum's inequality, and an old inequality of Minkowski and Radon.