Is the Godbersen conjecture true for mixed volume with reflection?

What the Godbersen inequality actually bounds

The object being maximized is not the volume of K itself but a mixed volume. Mixed volume V_n is the Minkowski-multilinear, symmetric polarization of volume, so when the difference body K−K is written as K+(−K), multilinearity expands V_n(K−K) into a sum of terms V_n(K[k], −K[n−k]) weighted by binomial coefficients C(n,k). The classical Rogers–Shephard inequality bounds the whole difference body by C(2n,n) V_n(K), with equality exactly for simplices. Godbersen in 1938, and independently Makai Jr. in 1974, conjectured that the same bound holds term by term, which is a strictly stronger statement because it constrains each mixed term separately rather than only their weighted sum.

The conjecture has two parts, and the paper settles both. The inequality part says V_n(K[k], −K[n−k]) ≤ C(n,k) V_n(K) for every 0 ≤ k ≤ n. The equality part says that when 1 ≤ k ≤ n−1 and K is full-dimensional, equality forces K to be a simplex. The endpoint cases k = 0 and k = n are trivial, since the mixed volume there is just V_n(K); lower-dimensional bodies make both sides zero. The cases k = 1 and k = n−1 already follow from the bound s(K) ≤ n on the Minkowski asymmetry measure. The open content is therefore the intermediate range of k, and it is exactly there that the paper's equality characterization does its work.

Why term-wise bounds are harder than the sum

A term-wise refinement is a priori a much riskier claim than the aggregate inequality it refines. The aggregate bound has one extremal family, but the family of term-wise inequalities could in principle admit many more extremizers, and inequalities involving mixed volumes are known to produce equality cases that are intricate to describe. The paper notes that it was nevertheless commonly expected that for full-dimensional K and 1 ≤ k ≤ n−1 the only equality cases would still be simplices, as recorded in standard references on convex geometry.

The paper's contribution is to confirm that expectation for general convex bodies, not only polytopes. Earlier work by the second- and third-named authors had proved the inequality for all convex bodies but characterized equality only among polytopes. The present paper closes that gap by characterizing equality for arbitrary convex bodies, and it does so through a structural result about simplices rather than through a direct case analysis of extremizers.

The monotonicity route to the inequality

The inequality is proved by embedding the problem into a product space and then applying monotonicity of mixed volumes. For convex bodies L ⊆ M and fixed C_1,…,C_{n−1}, monotonicity gives V_n(L, C_1,…,C_{n−1}) ≤ V_n(M, C_1,…,C_{n−1}). Writing K^2 = K×K as ι_1K + ι_2K and letting Δ be the diagonal embedding, one has ΔK ⊆ K^2, so the monotonicity inequality can be applied n times. Combined with the standard projection formula for mixed volumes, this yields V_n(K) V_n(K[k], −K[n−k]) ≤ C(n,k) V_n(K)^2, and dividing by the positive volume V_n(K) gives the Godbersen inequality.

The same argument scales to higher order. For p ≥ 1, the diagonal map Δ_p sends x to (x,…,x) in (R^n)^p, and K^p = ι_1K + ⋯ + ι_pK. The paper proves V_{pn}(−Δ_pK[n−k_0], ι_1K[n−k_1], …, ι_pK[n−k_p]) ≤ (n!)^p/(pn)! · C(n; k_0,…,k_p) · V_n(K)^p whenever k_0 + ⋯ + k_p = n. This is the higher-order Godbersen inequality, and it reduces to the original statement when p = 1. The inequality part is proved in this way; the harder half is deciding when equality can occur.

Equality becomes a condition on touching cones

Equality in the monotonicity inequality has a known reformulation in terms of the support of mixed area measures: if L ⊆ M and the relevant mixed volume with L equals that with M, then the support functions h_L and h_M agree on the support of a certain mixed area measure. Applying this to the product-space argument, an extremizer of the Godbersen inequality must satisfy h_{ΔK} = h_{K^2} on the support of S_{B[n−1], ι_1K[k], ι_2K[n−k]}.

The paper then converts this analytic condition into a geometric one using a local projection formula for mixed area measures and a known partial result on Schneider's conjecture about supports of mixed area measures. The outcome is Proposition 1.3: if a full-dimensional K attains equality, then for any two directions u, v on the unit sphere with dim T(K,u) ≤ k and dim T(K,v) ≤ n−k, there is a direction w such that the touching cone T(K,w) contains both T(K,u) and T(K,v). In words, equality forces the touching cones of K to be closed under a controlled join operation. This is the bridge from an inequality about volumes to a statement about the local normal structure of K.

A face-lattice characterization of simplices

Polarity turns the touching-cone condition into a condition on faces. If K contains the origin in its interior, touching cones of K correspond to faces of the polar body K°, so Proposition 1.3 becomes: for any two faces F and G of K° with dim F ≤ k−1 and dim G ≤ n−k−1, there is a proper face of K° containing both. The paper proves that this face-lattice condition already characterizes simplices, which is the technical heart of the equality analysis.

Theorem 1.4 states the equivalence of three conditions for a full-dimensional K and an integer 1 ≤ k ≤ n−1: K is a simplex; any two faces F, G with dim F ≤ k−1 and dim G ≤ n−k−1 lie in a common proper face; and any two directions u, v with dim T(K,u) ≤ k and dim T(K,v) ≤ n−k have touching cones contained in a common touching cone. The only previously known result in this direction was Develin's theorem for polytopes. The difficulty here is that the face lattice of a general convex body can be infinite, so the argument cannot enumerate faces. The proof instead works through an induction on k that weakens the dimension bound on one face while strengthening it on the other, using the fact that any proper face of a face has strictly smaller dimension.

Higher-order equality reduces to the two-face case

For the higher-order inequality the equality analysis follows the same arc. Proposition 1.7 says that if a full-dimensional K attains equality in the higher-order inequality, then for any p+1 directions u_0,…,u_p with dim T(K,u_i) ≤ k_i there is a single direction w whose touching cone contains every T(K,u_i). Theorem 1.8 extends the face-lattice characterization from two faces to p+1 faces: K is a simplex if and only if any faces F_0,…,F_p with dim F_i ≤ k_i−1 lie in a common proper face, and equivalently the corresponding touching-cone condition holds.

The reduction from p+1 faces to two faces is done by a lemma that merges two consecutive dimension bounds, repeatedly collapsing the list of constraints until only the two-face condition of Theorem 1.4 remains. The converse direction is short: if K is a simplex, its faces are themselves simplices, so faces with dim F_i ≤ k_i−1 together contain at most k_0 + ⋯ + k_p = n vertices, leaving a vertex outside all of them and hence a facet containing all the faces. This is why the equality cases of the higher-order inequality are again exactly the simplices, confirming Schneider's 2000 conjecture.

Where equality stops being trivial

The equality statement is deliberately restricted, and the restrictions matter for reading the result. If some k_i equals n, both sides of the higher-order inequality equal (n!)^p/(pn)! · V_n(K)^p, so equality always holds and the simplex characterization says nothing. If K is lower-dimensional, both sides are zero and equality is again trivial. Zero indices can be removed without changing whether equality holds, so the substantive case is k_0,…,k_p ≥ 1, which forces p ≤ n−1. The paper's equality characterization applies precisely in that regime, for full-dimensional K.

The practical consequence is a clean dichotomy rather than a numerical threshold. Within the nontrivial range, the extremal body is unique up to the stated conditions: a full-dimensional convex body attains equality in the Godbersen inequality, or in its higher-order form, exactly when it is a simplex. The paper's own framing places this alongside other characterizations of simplices as unique extremizers, such as the Minkowski asymmetry bound s(K) ≤ n, Ball's reverse isoperimetric inequality, Grünbaum's inequality, and Zhang's projection inequality. The distinctive contribution here is that the characterization is obtained through the face lattice, and that it covers general convex bodies rather than only polytopes.

What to take from the result

The most useful way to read this paper is as a statement about which structural feature controls equality. The inequality itself follows from monotonicity of mixed volumes applied in a product space, and that argument was already known to extend to higher order. What was missing was the equality analysis for general convex bodies, and the paper shows that the decisive information is not metric but combinatorial: equality forces a join-closure property on touching cones, which under polarity becomes a join-closure property on the face lattice, and that property alone pins the body down as a simplex.

For a reader comparing this to other simplex characterizations, the relevant distinction is the scope of the equality claim. The result covers full-dimensional convex bodies in the nontrivial index range 1 ≤ k ≤ n−1 for the classical inequality, and k_0,…,k_p ≥ 1 with sum n for the higher-order version, with the degenerate cases k_i = n and lower-dimensional bodies excluded because equality there is automatic. Within that range, the answer to the title question is unambiguous: the conjecture is true, and simplices are not merely extremizers but the only ones.

Sources used in this answer