Anisotropy in arbitrary characteristic, Lefschetz beyond positivity, moment curves and lattice polytopes (part 2)

Just a quick post that kills some questions from the previous post, and a report on just an overall wonderful REU project summer.

So let me start by the following: myself, and then Papadakis and Petrotou, and then us three jointly, proved the Lefschetz property for triangulated spheres (characteristic two in the case of Papadakis and Petrotou). This may be an unfamiliar word for someone not from algebraic geometry, but essentially, it is a property stating that Poincaré duality in certain manifolds coming from algebraic geometry (for instance, smooth projective varieties) is realized in a concrete way. Now you may think combinatorial Hodge theory à la Rota conjecture I proved with June and Eric, or positivity of Kazhdan-Lustig polynomials established by Elias and Williamson, this is much better. Because while those arguments relied on a known combinatorial trick by McMullen/de Cataldo-Migliorini, this one used an entirely new idea. First, let me state the theorem, without getting too technical:

We proved, given a triangulated sphere \Sigma of dimension d-1, the face ring (in arbitrary characteristic) permits an Artinian reduction and contains a linear element \ell so that

A^k(\Sigma) \xrightarrow{\ell^{d-2k}} A^{d-k}(\Sigma)

is an isomorphism.

Too explain some terms: Initially the face ring is too large to permit such a statement. It is only after an Artinian reduction, parametrized by choosing d linear forms, that the statement is true. And it is far beyond what algebraic geometers know and can prove, cool.

We generalized this to cycles later, meaning we actually obtained a Lefschetz in probably the biggest generality one could imagine in this setting. This implied all kinds of wonderful things. In particular, it implied the Grünbaum conjecture for simplicial complexes embeddable in some manifold. For instance, if
X is a simplicial complex that PL embeds into \mathbb{R}^{2k}, then the number of k dimensional simplices is at most (k+2) times more than the number of k-1-simplices.

Now, there were several questions left, some of which are now solved.

Restricting the Artinian reduction The probably most exciting thing first: the result of myself, Papadakis and Petrotou only implied the Lefschetz property in a “generic” Artinian reduction. We could not say much more about how to choose the Artinian reduction explicitly, but for a program I have regarding the Hopf and Charney-Davis conjectures, I needed to restrict further.

So, this summer, I had four REU students over in wonderful Jerusalem. Kaiying Hou (Harvard) and Daishi Kiyohara (MIT) and Daniel Koizumi (Utah) and Monroe Stephenson (Portland). I suggested putting a restriction on the types of Artinian reductions. And I had a bold conjecture that I only ever made to be provocative: we can choose points on the moment curve: If our sphere has n vertices, then we can choose t_i n points on the moment curve such that the the linear forms \theta_j= \sum t_i^j x_i, j ranging from 0 to d form a legit linear system for our face ring. You cannot choose a lower degree curve to achieve this. So no hope for Lefschetz, I thought. But was proven wrong. After Daishi solved this problem in characteristic two and zero, we (The REU crew) obtained

Theorem The Artinian reduction induced by the moment curve implies the Lefschetz property.

Daishi, Daniel, Monroe and Kaiying sitting on top of Jerusalem’s old city

Moment curves are just cool, and this result reflects a recent result of Luis Crespo Ruiz and Paco Santos, which essentially states that generic rigidity does not have to be generic, just moment curve general (I explained the connection between Lefschetz theory and rigidity earlier, remember?). Seems moment curves have got it all.

Anisotropy and reduced rings Now one key to the argument for the Lefschetz property is to understand how the Poincaré pairing restricts to subspaces. Recently I posted a somewhat simplified argument for the Lefschetz theorem, and in fact it led to another theorem I had long conjectured. First, Daishi and I proved the following theorem for m=p the characteristic of the underlying field, but then a general version was quickly obtained

Theorem Consider K any field, \mu any (d-1)-dimensional cycle over K, and the associated graded commutative face ring. Then, for some field extension K’ of K, we have an Artinian reduction B(\mu) that is almost reduced, i.e. for every element u\in B^k(\mu), $k\le \frac{d}{m}$, we have

u^m\ \neq\ 0

That is just cool. Finally, I want to discuss lattice polytopes some more, and make up for the problems I solved with a more general one.

The curious case of Benjamin Button lattice polytopes Now all of these proofs and resultsare rather combinatorial. And what better than combinatorics? Arithmetic. A little at least. So any lattice polytope generates a semigroup algebra. And sometimes this algebra is standard. In this case, we (Papadakis, Petrotou, Steinmeyer and myself) prove the Lefschetz property (with the above trick, finally in general characteristic) for standard (= IDP) lattice polytopes, and solve just a bunch of conjectures. So now it remains to understand when this whole trick can work in general. So let me formulate a conjecture and call it a problem:

Problem Consider a commutative standard (generated in degree 1) graded algebra which is Gorenstein, its Krull dimension is at least one and finally, it is reduced (no nilpotent elements). Then it has an Artinian reduction that is Lefschetz and almost reduced.

Scalar curvature and angles of polyhedra: test your intuition

I was listening to a marvelous talk by Jean-Pierre Bourguignon today, about generalizations and (geo)metric meaning of scalar curvature, and Misha Gromovs thoughts on the subject. Essentially, just like Alexandrov and Gromov and many others gave us metric understanding of sectional curvature, and Sturm and Lott-Villani and many more gave us metric ways of thinking about Ricci curvature, Misha is at it again to do the same with the weakest notion of curvature.

One of the goals would be to understand polyhedral spaces from the viewpoint of scalar curvature. In this context, Misha asked me to give a reference for the following statement: Given fixed d and any \epsilon>0, there is a finite number of combinatorially distinct d-polytopes all whose dihedral angles are all smaller than \pi-\epsilon. I will write more about this and scalar curvature later, but for now I am stealing a format from Gil’s blog and ask: Did I have a chance to find such a reference, or do you find examples that make such a statement hopeless? And what \epsilon>0 can you choose safely?

While I am at it, let me add another question for your intuition: Can you deform a polytope such that all dihedral angles do not get bigger? What deformations can you find? More on this subject, and some partial answers to the questions above, later.

Called her Jiji

A note on PL handlebodies, the Hausmann trick and some homology spheres

There is a surprising lack of intuition for PL manifolds around, which always surprised me. And it turns out you can answer some questions. We stay in the category of PL manifolds throughout. The following question was asked by Gil Kalai to Ed Swartz some 15 years ago. Ed could not answer the question, and popularized here.


Question (Kalai) Are there different (homeomorphism types of) triangulated closed compact homology (d-1)-spheres $H$ with g_3=0 (for any d)?

Without going too deep into what g_3=0 means, we note quite simply that it is equivalent to saying that H is boundary to a triangulated manifold D that has no interior simplices of dimension d-3.

The answer to this is yes. In fact, there are infinitely many (). For this purpose, we will make the following observation:

Theorem H. Let d \ge 2. Let M be a PL d-manifold. The following are equivalent:

1. M admits a PL handle decomposition into handles of index \le k,

2. M admits a PL triangulation in which all (d-k-1)-simplices are on \partial M.

Both directions are quite easy, and probably folklore; we shall need the implication from 1. to 2. here, for the converse, consider the dual of the triangulation, and attach the cells one by one in order of increasing dimension. Ed conjectured the implication from 2. to 1., but interestingly seemed to believe that the answer to Gil’s question is no. Alas, this theorem implies that the answer to Gil’s question is yes.

The Hausmann trick So what we want is an infinite number of homology spheres that are

  1. Boundary to homology disks D constructed only from 0, 1, and 2 handles
  2. Are of distinct homeomorphism type. We use their fundamental groups as invariants.

To this end, let us construct D first. They are obtained as d-dimensional handlebodies, d\ge 6 , constructed over twodimensional presentation complexes for distinct perfect groups with balanced presentations (meaning number of generators = number of relations).

This takes care of both requirements: First, note that we only use handles corresponding to cells of the presentation complex; these are of dimension two only.

Second, \pi_1(\partial D)= \pi_1(D) by general position in dimension principle in dimension 6 (the presentation complex, perturbed into general position, does not intersect itself and therefore can be homotoped to the boundary.)

So how do we get a perfect group with a balanced presentation? This is simple, and was taught to me by my friend Louis Funar: pick any finitely generated perfect group, and write its generators in terms of commutators, one for each: This obviously provides a perfect group again, a priori larger, but it has balanced presentation. Now, we can simply consider the direct product of such groups to get infinitely many. It remains to provide the

Proof of Theorem H We need a lemma

Lemma G. Let k \ge 1 be an integer. Let C be a PL-embedded (k-1)-complex in a PL (d-1)-complex N, where k\le d. Let T be a PL triangulation of N such that C is transversal to T (j faces intersect i faces in dimension $i+j-d+1$, or not at all). Then there is a subdivision T' of T such that

1. the neighborhood of C inside T' is regular, and C is transversal to $T’$;

2. T' is obtained from T by stellarly subdividing only faces of dimension \ge d-k.

Here, stellar subdivision is the subdivision obtained by removing a face, and coning over the boundary of the hole left; we can think of it as simply taking the cone over the neighborhood of the face in question, and forgetting about the part covered by it. In particular, if N=\partial M, then this operation of stellarly subdividing at a face of dimension s introduces an interior face of dimension $s$, but not of lower dimension.

Regular neighborhood means that the collection of faces incident to C strongly PL deformation retracts onto C. For this, it is enough to show that the new subdivision T' has the following property:

If v is any vertex of T', then the restriction of $C$ to the neighborhood (the star) st_v N of that vertex of T' is conical (it has a unique minimal face), and hence contractible.

For this, we can use the following observation: If P is any polyhedron, and A, B are disjoint polytopes inside it so that they intersect no common facet of $P$, then they can be seperated using stellar subdivisions at maximal faces. This is trivial, as a single such subdivision suffices up to PL homeomorphism.

We can now prove this statement by induction on the dimension of N, assuming by induction that we have proven the statement for the codimension one skeleton of N, and the restriction of $C$ to that skeleton.

Now, given N, we therefore may assume that the statement holds for the boundary of st_v N, and apply the observation. This finishes the proof of the lemma.

Now we can finish the proof of Theorem H, which we do by induction on dimension of the manifold. For this, we attach the handles one by one. In a single step, say from M_\ell to M_{\ell+1}, consider the attaching sphere S for the handle PL embedded into \partial M_\ell. By Lemma G we may assume that the neighborhood U of S is regular. Consider K
the cone over U. This is not a manifold, but we can turn it into one by considering the tangent space at the conepoint. By induction on the dimension we can turn the link into a disk K' without adding interior faces whose codimension exceeds the index of the handle. Set M_{\ell+1}=M_{\ell} \cup K' That finishes the proof. See also here.

Sofia, Lefschetz via shelling and bonk and the holy trinity (kind of) (part 1)

This is born out of an attempt to find equivariant Lefschetz elements, so to have a combinatorial Lefschetz theorem that is a little less generic for some conjectures in geometric topology. Actually this succeeds to give some interesting results, and I will update in a second part. It will take me some days, in the meantime I made this here simple and clear.

Dear X,

so, Covid seems almost over (fingers crossed). I lost three I loved during the time (none of them due to covid, funny enough; two suicides and a heart failure), I had covid twice (fingers crossed for the hat trick) despite three vaccinations. I am in Sofia, at a inaugural conference of the ICMS.

But that’s not what this post is about. This post is about another proof of the g-conjecture. Well, the Lefschetz property for simplicial cycles, really. It is the simplest one yet, but that is not why it is important. It is also the third one (that is essentially different) and combines the ideas of three teams. (I count the original one by me here, and the characteristic two proof by Papadakis and Petrotou; our joint paper is a combination of the ideas from the former and yada yada yada creative counting to make the holy number work out. Deal with it.) If the first is a refined choreography of slashes and parries that is difficult to follow, and the second the equivalent of wooshing around with bloodhound step (using a miraculous formula that comes out of nowhere), then this is the equivalent of bonking the boss with a hammer: we write down a rational function and examine it, observing it has a pole to show it is nontrivial. Unfortunately it seems to be less general than either of the previous proofs, but I will see whether it can be pushed.

So I assume that I have a PL sphere \Sigma of even dimension 2k-2.

I also restrict myself to the middle Lefschetz property. That means that I want to prove, for some Artinian reduction A(\Sigma)[\Theta] of the face ring \mathbf{k}(\Sigma), \mathbf{k} any field you like, with respect to the linear system of parameters \Theta, that we have an isomorphism.

A^{k-1}(\Sigma)[\Theta] \xrightarrow{\cdot \ell} A^{k}(\Sigma)[\Theta]

induced by multiplication with an element \ell. Finally, this proof also only seems to work in characteristic two and zero (update: it works in general char, but is more involved. Wait for part 2)

Those are all restrictions I make here, the full argument I will write in a note soon.

Now, remember that cool property called biased pairings I told you about? So, if D is a disk of dimension d-1, and 2i\leq d, then there is the Poincaré pairing

A^i(D) \times A^{d-i}(D, \partial D)\rightarrow A^{d}(D, \partial D) \cong \mathbf{k}.

Now, as you remember, if \partial D=\Sigma, then the middle Lefschetz property is equivalent to saying that the pairing

A^k(D,\Sigma)[\Theta,\ell] \times A^{k}(D, \partial D)[\Theta,\ell]\rightarrow A^{d}(D, \partial D)[\Theta,\ell] \cong \mathbf{k}

is perfect (which I defined using the map A^k(D,\Sigma)\rightarrow A^k(D) ). This is the biased pairing property I discovered for my original proof, and it was the breakthrough needed to prove the Lefschetz property in the combinatorial setting. Indeed, all proofs rely on it, and it is my great pride and shame. Pride because it was then rediscovered by Kalle Karu and Stavros Papadakis and Vasso Petrotou, which affirmed it was the right way of going about the problem. Shame because “rediscovered” means that it was written in such a difficult way that they had to reinvent it.

For future reference, let us denote the identification

A^{d}(D, \partial D) \cong \mathbf{k} by deg. We will choose a natural normalization later.

Anyway, to continue in the text: One of the ideas to prove the biased pairing property is to use a simple decomposition of the disk D. In my original paper, I used a decomposition that was rather involved, but Kalle Karu had the ingenious idea of using just a shelling of D. His proof did not quite work, but I will avenge him here.

A shelling is a way of adding facets (simplices of dimension d-1) one by one so that I stay a disk (D_i) at any point in time. In fact, I can assume that D is shellable by a classical result of Pachner: Every PL sphere is the boundary of a shellable disk.

Now, we can actually give a basis of A^k(D,\Sigma)[\Theta,\ell] along such a shelling (D_i): a new monomial is added whenever there is a new cardinality k simplex \tau_j in (D_i, \partial D_i) whose boundary \partial \tau_j lies in \partial D_i. Here j is just an index that rises whenever such a face occurs. Notice that in such a case, a facet F_j is added to (D_i) that has not appeared in any of the previous \tau_s,\ s<j

That is the second trick. Now we turn the corresponding monomials x_j=x_{\tau_j} into an orthogonal basis A^k(D,\Sigma)[\Theta,\ell] under the Poincaré pairing using the Gram-Schmidt process. Let us denote the orthogonalization of x_j with respect to $ x_s,\ s<j by \pi_j.

Here, the final team and its idea enters: Stavros Papadakis and Vasso Petrotou suggested we look function fields, and hence not at the field \mathbf{k}, but at the field of rational functions over \mathbf{k} with indeterminates the coeffcients of \Theta'=(\Theta,\ell). With this, for instance, it is natural to choose, for F a maximal face of (D,\Sigma)

deg(x_F)= sgn(F)\cdot (det(\Theta'_{|F}))^{-1}

Here sgn(F) is the sign of F with respect to some chosen ordering on the vertices, and \Theta'_{|F} is the minor of the matrix \Theta' given by the columns corresponding to F.

For monomials that are not squarefree, the degree can be computed as well, as observed in the following formula of Carl Lee:

deg (\mathbf{x}^\alpha)(\Theta')\ =\ \sum_{F \text{ facet containing } \mathrm{supp}\ \alpha } \deg(\mathbf{x}_{F}) \prod_{i\in \mathrm{supp}\ F} (det(\Theta'_{|F-i}))^{\alpha_i-1}

where we understand \Theta'_{|F-i} to be the matrix obtained by replacing the i-th column of \Theta'_{|F} with any fixed vector.

The key is now to understand


Recall that \pi_j=x_j-proj_{x_s,\ s<j}(x_j). Hence

deg(\pi_j^2)=deg((x_j-proj_{x_s,\ s<j}(x_j) )^2)

We can expand this and compute with Lee’s formula: Notice now that the only time the determinant of the minor of \Theta'_{|F_j} appears is in the expansion of deg(x_j^2), where its inverse appears. (As Kaiying Hou pointed out to me, I should remind everyone here that we restrict to characteristic 2 here, so that the mixed term in the square vanishes. Then we can assume the remaining term does not have a pole by induction on j. Thanks Kaiying)

To make this really simple, consider what happens if we have k vertices of D and move them around. Formally, pick those vertices, and consider the partial differentials of moving them into k linearly independent directions.

Claim: The composition of these partial differentials on deg(\pi_j^2) vanishes unless the vertices form a face of D. It is nonzero on F_j - \tau_j

The first part is a simple induction. The second a simple calculation.

Hence, keeping everything else in general position but degenerating that minor, the deg(\pi_j^2) has a pole, so the rational function is not trivial. But this means that the orthogonalization leads to a matrix for the biased pairing that is diagonal with nontrivial diagonal entries deg(\pi_j^2). Hence the matrix is nondegenerate, and the pairing perfect. Done.

Concluding Remarks

Well why did we make this effort? For that you will have to be a little patient. In short, it turns out that this new method allows for combinatorial Lefschetz elements that are more restricted than just generic, and in particular those that are equivariant under certain group actions. Why is that important? Well, the key lies in the book of Gromov, psalm 137.

An intriguing question that remains open is to show full anisotropy for general simplicial cycles. I know exploited that I could show biased pairing, that is, nondegeneracy of the Poincaré pairing at an ideal (namely A^k(D,\Sigma)). But Papadakis and Petrotou (and then with me) showed in characteristic two the Poincaré pairing degenerates at no ideal in a Gorenstein ring; that seems more tricky. I will post some more detailed conjectures soon, and go into quadratic forms on function fields. Cool stuff.

Finally let me announce a theorem that is actually new, and is joint work with Stavros Papadakis, (my future postdoc) Vasso Petrotou and (my current student) Johanna Steinmeyer: we are now able to push the Lefschetz principle beyond combinatorial settings (using yet another method), and prove in particular

Theorem The h^\ast vector of an IDP Gorenstein lattice polytope is unimodal.

See you all soon, Love, Karim

Prague and overdue congratulations

I visited Prague last week to distract myself after my cat Misha died way too early (he was only 6). And it was quite an amazing visit, in every way. Something about it made me forget my worries (can you guess what?)

More importantly, I met once again my very first postdoc Zuzka (Patakova) as well as my good friend Martin (Tancer). As is long-establish precedent among postdocs, Zuzka had to share some of her blueberry cake with me. But we also made some amazing advances, proving that realizing simplicial 4-polytopes is at least as hard as existential theory of the reals as well as investigating cool questions about spaces with prescribed geodesics.

Let me close with some overdue congratulations: My former postdoc Gaku Liu (now a professor in University of Washington) extended some joint work of him, Michael Temkin and me to prove an old conjecture in polytope theory: Every sufficiently large dilation of a lattice polytope admits a unimodular triangulation.

Oh and my coauthor Vasso has succesfully defended her thesis. She will be joining my team in the fall to work on my new ERC project. Congratulations Gaku and Vasso!

Just when you think it’s over

Igor Pak's blog

The past is never dead. It’s not even past,” memorably wrote William Faulkner. He was right. You really have to give the past some credit — it’s everlasting and all consuming. Just when you think it’s all buried, it keeps coming back like a plague, in the most disturbing way.

The story here is about antisemitism in academia. These days, in my professional life as a mathematician, I rarely get to think about it. As it happens, I’ve written about antisemitic practices in academia and what happened to me on this blog before, and I didn’t plan to revisit the issue. After thirty years of not having to deal with that I was ready let it go… Until today. But let me start slowly.

The symbolism

In American universities, the antisemitism was widespread practice for decades which went out of fashion along with slide rule and French curve

View original post 674 more words

Degrees, curious identities, Lefschetz maps and biased pairings (and a cliffhanger)

This post is long overdue, but being in the middle of nowhere helps finally progressing a little.

In the last post, I discussed a lemma of Kronecker. Let me discuss another version

Lemma (Kronecker, second version) Consider A, B linear maps from a vectorspace X to another one Y over any infinite field. Assume that im(A) and B ker(A) intersect trivially. Then the generic linear combination of A and B has kernel equal to the intersection of kernels of A and B, in other words, the kernel is as small as possible.

This is a very powerful tool to construct high-rank maps in vector spaces, in particular in the case when X and Y are dual in a Poincaré duality algebra. Then im A and ker A are orthogonal complements, and linear algebra tells us that orthogonal complements intersect trivially if and only if the Poincaré pairing does not degenerate on either space. This is exploited in my proof of the g-conjecture to construct Lefschetz elements, see here, and also in this survey (which I was invited to for winning the EMS prize, which I still cannot quite believe).

I call this property, that a pairing does not degenerate on certain subspaces, the biased pairing property. The relevance of biased pairings to the Lefschetz property was then rediscovered by two Greek mathematicians, Stavros Papadakis and Vasiliki Petrotou, who gave a marvellous second proof of the g-conjecture, though initially only in characteristic two.

They proved stronger that in a certain field extension (and characteristic two), the pairing does not degenerate at ANY ideal (let’s call this total anisotropy). This is quite marvellous. In essence, they proved that if the Artinian reduction of the equivariant cohomology ring of the variety is chosen along indendent transcendental variables, then the fundamental class deg satisfies

\partial_\sigma deg (u^2) = deg (x_\sigma u)^2

where \partial_\sigma is a partial differential after the transcental variables in question, specifically in direction of vertices of a simplex \sigma. Since the right-hand has to be non-zero for SOME simplex \sigma. Hence deg (u^2) is nonzero, and the biased pairing property holds for every principal ideal, and in particular any ideal.

We later extended this partially to general characteristic, providing a second complete proof of the g-conjecture, see here. So far, so good. You can see a detailed explanation in the Hadamard Lectures I was honored to give, here.

Anisotropy and biased pairings seems to be of great importance in generic algebraic geometry going forward, and I have been wondering how far this extends, in particular the identity above and biased pairings. In any case, you can count on some exciting news in this direction soon.

Schmidt and analytic ranks, the Kronecker lemma and Hadamard lectures

Just a short post: David Kazhdan, Tammy Ziegler and I have recently (and finally) posted our proof that the analytic rank of a cubic, the logarithm of its bias, and its Schmidt rank (how hard it is to write it as products of lower degree polynomials), are linearly related.

This came out of our discussion of Kronecker’s lemma. One version can be stated as follows:

Lemma (Kronecker) Consider A, B linear maps from a vectorspace X to another one Y over any infinite field. Assume that

B ker(A) is not a subspace of im A

then the generic linear combination of A and B has larger rank than A and B

Leopold Kronecker

I had used this lemma in a previous work to construct Lefschetz elements, without knowing at first it was due to Kronecker. More about that Lefschetz aspect later, when some exciting news will come online. Suffices to say, that Kronecker’s lemma gives a rather cool restriction on spaces of linear maps, and strongly restricts them when they are of low rank.

Corollary Consider a space of bilinear forms L over X times Y, such that every element is of rank at most r. Then there are subspaces X’ and Y’, each of codimension r in X resp. Y, so that L vanishes identically on X’ times Y’.

It isn’t much, really, but at first this corollary was quite counterintuitive to me, since after knowing that a space of matrices of rank r cannot be restricted to r rows or r columns (consider \mathfrak{so}_3) I did not believe anything of the sort could be true.

Anyway, for the application to tensor ranks, see our paper. Let me advertize also the upcoming Hadamard lectures, where I will explain how Kronecker’s lemma can be used to construct Lefschetz isomorphisms.

(including a picture of myself where I look like a mobster)

Kronecker, L. Algebraische Reduction der Scharen bilinearer Formen, Sitzungsber. Akad.Berlin, Jbuch. 22 (169) (1890) 1225-1237.

Minkowski sums (and how big they must be)

So, this is the promised actual first post, motivated by and answering a question recently posed to me by Yue Ren (and several before him that I apologize for forgetting). It comes with a problem, and that is, to prove the result in a simpler, more elementary way.

I had written a longer paper (with Raman Sanyal) on a problem concerning Upper Bounds on the complexity of Minkowski sums that dealt with the problem quite thoroughly.

But after we submitted the paper, and it was accepted and published in Publications IHES, someone asked me a question. I thought for a while, and answered that it follows from that and that lemma in the paper. But the question came up again and again, seemingly being quite relevant. On the other hand, writing a new paper on the matter was tedious, as it promised to be 90% repetition of paper number one. Which was an annoying prospect at best.

The problem is this: Consider a bunch (say m of polytopes (P_i) in a euclidean vector space of dimension d. For simplicity, and to be interesting, let us assume that all these polytopes are of positive dimension. Also, let us assume that all these polytopes are in general position with respect to each other.

Question: How many vertices must the Minkowski sum of the family (P_i) have? Surely, it has to be at least the number of vertices as the Minkowski sum of m segments in general position.

Turns out, the naive guess is correct. Also turns out that the proof seems to be quite a bit harder than I initially thought it would be. Indeed, the naive idea, that the Minkowski sum of two polytopes of positive dimension P and Q, P fixed, is minimized if and only if Q is a line segments, is not true. The real proof, unfortunately, seems to need a considerable detour. Let me sketch it here.

The first idea here is to consider the Cayley polytope of the family (P_i). This is a polytope in \mathbb{R}^{d+m-1}= \mathbb{R}^{d} \otimes \mathbb{R}^{m-1} obtained by translating each of the P_i independently into general positon, for instance by translating each of them along a choice of m vectors v_i in general position in \mathbb{R}^{m-1} that sum to zero. And then passing to the convex hull.

That is a rather powerful trick, as we now have a new polytope C(P_i) that encodes the Minkowski sum: we obtain it back by slicing the polytope with the subspace \mathbb{R}^{d}. The collection of faces of C(P_i) that intersects \mathbb{R}^{d} is called the “open Cayley complex” T^\circ(P_i), its closure is the Cayley complex T(P_i).

Now we count the number of faces f_j of the Minkowski sum, or equivaently, the face numbers of the open Cayley complex (shifted by m-1). That is best done by first transforming to the h-vector,

h_k := \sum_{i=0}^k (-1)^{k-j}\binom{d-j}{k-j}f_{j-1}.

and estimating this. There are now two essential facts: We have the Dehn-Sommerville relations

h_k(T(P_i)) \ =\ h_{d+m-j}(T^\circ(P_i)) + E

where the error E only depends on number of summands, and the important inequality

h_1(T(P_i)) \ \le\ h_{d-j}(T^\circ(P_i))

for all j\ge 1. That last inequality is key, and seems to only really follow from that Lefschetz theorem for polytopes (or the associate toric variety) where the first, the Dehn-Sommerville relations, follow from “only” a version of Poincar\’e duality.

Now the path is clear: if we want to estimate the number of k-faces of the Minkowski sum, then we can equally estimate the number of k+m-1-faces of the open Cayley complexes, which amounts to estimating h_{k+m}(T^\circ(P_i)), or equivalently, h_{d-k+1}(T^\circ(P_i)).

Which of course is bounded below by h_1(T(P_i)), which is quickly computed to be the total number of vertices of summands, minus d+m-1, and gives the desired lower bound.

The details can be found here and in its appendix here.