The first two coefficients in Wills' lattice point inequality

For a convex body K ⊂ ℝⁿ let G(K) be the number of integer points in K, and let V₀(K), …, Vₙ(K) be its intrinsic volumes. Wills conjectured in 1973 that G(K) ≤ V₀(K) + ⋯ + Vₙ(K). This holds for n ≤ 3 but fails in high dimensions, and Henk asked whether at least the coefficients of Vₙ and Vₙ₋₁ are correct. We show that they are: for every n there is a constant Cₙ such that G(K) ≤ Vₙ(K) + Vₙ₋₁(K) + Cₙ(V₀(K) + ⋯ + Vₙ₋₂(K)) for all convex bodies K ⊂ ℝⁿ. The proof is an induction on the dimension, slicing by the hyperplanes xₙ = k. Its main step bounds the error of the Riemann sum of the slice volumes of a polytope whose vertices have integral heights. The bound comes from an exact decomposition of this error into contributions of facets and of faces of codimension two.