Simultaneously generalizing both neighborly and neighborly cubical polytopes, we introduce PSN polytopes: their -skeleton is combinatorially equivalent to that of a product of simplices.
We construct PSN polytopes by three different methods, the most versatile of which is an extension of Sanyal & Ziegler’s “projecting deformed products” construction to products of arbitrary simple polytopes. For general and , the lowest dimension we achieve is .
Using topological obstructions similar to those introduced by Sanyal to bound the number of vertices of Minkowski sums, we show that this dimension is minimal if we additionally require that the PSN polytope is obtained as a projection of a polytope that is combinatorially equivalent to the product of simplices, when the dimensions of these simplices are all large compared to .
Let denote the -dimensional simplex. For any tuple of integers, we denote by the product of simplices . This is a polytope of dimension , whose non-empty faces are obtained as products of non-empty faces of the simplices . For example, Figure 1 represents the graphs of , for .
We are interested in polytopes with the same “initial” structure as these products.
Let and , with and for all . A convex polytope in some Euclidean space is -prodsimplicial-neighborly — or -PSN for short — if its -skeleton is combinatorially equivalent to that of .
We choose the term “prodsimplicial” to shorten “product simplicial”. This definition is essentially motivated by two particular classes of PSN polytopes:
neighborly polytopes arise when ;
neighborly cubical polytopes [js-ncps-05, sz-capdd] arise when .
In the literature, a polytope is -neighborly if any subset of at most of its vertices forms a face. Observe that such a polytope is -PSN with our notation.
The product is a -PSN polytope of dimension , for each with . We are naturally interested in finding -PSN polytopes of smaller dimensions. For example, the cyclic polytope is a -PSN polytope of dimension . We denote by the smallest possible dimension of a -PSN polytope.
PSN polytopes can be obtained by projecting the product , or a combinatorially equivalent polytope, onto a smaller subspace. For example, the cyclic polytope (just like any polytope with vertices) can be seen as a projection of the simplex to .
A -PSN polytope is -projected-prodsimplicial-neighborly — or -PPSN for short — if it is a projection of a polytope that is combinatorially equivalent to .
We denote by the smallest possible dimension of a -PPSN polytope.
1.2. Outline and main results
The present paper may be naturally divided into two parts. In the first part, we present three methods for constructing low-dimensional PPSN polytopes:
Reflections of cyclic polytopes;
Minkowski sums of cyclic polytopes;
Deformed Product constructions in the spirit of Sanyal & Ziegler [z-ppp-04, sz-capdd].
The second part derives topological obstructions for the existence of such objects, using techniques developed by Sanyal in [s-tovnms-09] (see also [rs-npps]) to bound the number of vertices of Minkowski sums. In view of these obstructions, our constructions in the first part turn out to be optimal for a wide range of parameters.
We devote the remainder of the introduction to highlighting our most relevant results. To facilitate the navigation in the article, we label each result by the number it actually receives later on.
Our first non-trivial example is a -PSN polytope in dimension , obtained by reflecting the cyclic polytope through a well-chosen hyperplane:
Proposition 2.3. For any , and sufficiently large, the polytope
is a -PSN polytope of dimension .
For example, this provides us with a -dimensional polytope whose graph is the cartesian product , for any .
Next, forming a well-chosen Minkowski sum of cyclic polytopes yields explicit coordinates for -PPSN polytopes:
Theorem 2.6. Let and with and for all . There exist index sets , with for all , such that the polytope
is -PPSN, where Consequently,
For we recover neighborly polytopes.
Finally, we extend Sanyal & Ziegler’s technique of “projecting deformed products of polygons” [z-ppp-04, sz-capdd] to products of arbitrary simple polytopes: given a polytope that is combinatorially equivalent to a product of simple polytopes, we exhibit a suitable projection that preserves the complete -skeleton of . More concretely, we describe how to use colorings of the graphs of the polar polytopes of the factors in the product to raise the dimension of the preserved skeleton. The basic version of this technique yields the following result:
Proposition 3.4. Let be simple polytopes of respective dimension , and with many facets. Let denote the chromatic number of the graph of the polar polytope . For a fixed integer , let be maximal such that . Then there exists a -dimensional polytope whose -skeleton is combinatorially equivalent to that of the product provided
A family of polytopes that minimize the last summand are products of even polytopes (all 2-dimensional faces have an even number of vertices). See Example 3.5 for the details, and the end of Section 3.1 for extensions of this technique.
Specializing the factors to simplices provides another construction of PPSN polytopes. When some of these simplices are small compared to , this technique in fact yields our best examples of PPSN polytopes:
Theorem 3.8. For any and with ,
where is maximal such that .
If for all , we recover the neighborly cubical polytopes of [sz-capdd].
In order to derive lower bounds on the minimal dimension that a -PPSN polytope can have, we apply and extend a method due to Sanyal [s-tovnms-09]. For any projection which preserves the -skeleton of , we use Gale duality to construct a simplicial complex that can be embedded in a certain dimension. The argument is then a topological obstruction based on Sarkaria’s criterion for the embeddability of a simplicial complex in terms of colorings of Kneser graphs [m-ubut-03]. We obtain the following result:
Corollary 4.13. Let with .
If then .
Let with and for all . For any such that , the smallest -PPSN polytope has dimension exactly . In other words:
During the final stages of completing this paper, we learned that Rörig and Sanyal [rs-npps] also applied Sanyal’s topological obstruction method to derive lower bounds on the target dimension of a projection preserving skeleta of different kind of products (products of polygons, products of simplices, and wedge products of polytopes). In particular, for a product of identical simplices, , they obtain our Theorem 4.9 and a result (their Theorem 4.5) that is only slightly weaker than Theorem 4.12 in this setting (compare with Sections 4.5 and 4.6).
2. Constructions from cyclic polytopes
Let be the moment curve in , be distinct real numbers and denote the cyclic polytope in its realization on the moment curve. We refer to [z-lp-95, Theorem 0.7] and [lrs-tri, Corollary 6.1.9] for combinatorial properties of , in particular Gale’s Evenness Criterion which characterizes the index sets of upper and lower facets of .
Cyclic polytopes yield our first examples of PSN polytopes:
For any integers and , the cyclic polytope is -PPSN.
For any and with and for all , define . Then the product
is a -PPSN polytope of dimension (which is smaller than when is nonempty). Consequently,
2.1. Reflections of cyclic polytopes
Our next example deals with the special case of the product of a segment with a simplex. Using products of cyclic polytopes as in Example 2.2, we can realize the -skeleton of this polytope in dimension . We can lower this dimension by by reflecting the cyclic polytope through a well-chosen hyperplane:
For any , and sufficiently large, the polytope
is a -PSN polytope of dimension .
The polytope is obtained as the convex hull of two copies of the cyclic polytope . The first one lies on the moment curve , while the second one is obtained as a reflection of with respect to a hyperplane that is orthogonal to the last coordinate vector and sufficiently far away. During this process,
we destroy all the faces of only contained in upper facets of ;
we create prisms over faces of that lie in at least one upper and one lower facet of . In other words, we create prisms over the faces of strictly preserved under the orthogonal projection with kernel .
The projected polytope is nothing but the cyclic polytope . Since this polytope is -neighborly, any face of dimension at most in is strictly preserved by . Thus, we take the prism over all faces of of dimension at most .
Thus, in order to complete the proof that the -skeleton of is that of , it is enough to show that any -face of remains in . This is obviously the case if this -face is also a -face of , and follows from the next combinatorial lemma otherwise. ∎
A -face of which is not a -face of is only contained in lower facets of .
Let be a -face of . We assume that is contained in at least one upper facet of . Since the size of the final block of an upper facet of a cyclic polytope is odd, contains . If , then is a facet of containing . Otherwise, , and has only elements. Thus, is a face of , and can be completed to a facet of . Adding the index back to this facet, we obtain a facet of containing . In both cases, we have shown that is a -face of . ∎
2.2. Minkowski sums of cyclic polytopes
Our next examples are Minkowski sums of cyclic polytopes. We first describe an easy construction that avoids all technicalities, but only yields -PPSN polytopes in dimension . After that, we show how to reduce the dimension to , which according to Corollary 4.13 is best possible for large ’s.
Let and with and for all . For any pairwise disjoint index sets , with for all , the polytope
is -PPSN, where
The vertex set of is indexed by . Let define a -face of . Consider the polynomial
Since indexes a -face of , we know that , so that the degree of is indeed . Since , and equality holds if and only if , the inner product equals
with equality if and only if . Thus, indexes a face of defined by the linear inequality .
We thus obtain that the -skeleton of completely contains the -skeleton of . Since is furthermore a projection of , the faces of are the only candidates to be faces of . We conclude that the -skeleton of is actually combinatorially equivalent to that of . ∎
To realize the -skeleton of even in dimension , we slightly modify this construction in the following way.
Let and with and for all . There exist pairwise disjoint index sets , with for all , such that the polytope
is -PPSN, where
We will choose the index sets to be sufficiently separated in a sense that will be made explicit later in the proof. For each -face of , indexed by , our choice of the ’s will ensure the existence of a monic polynomial
which, for all , can be decomposed as
where is an everywhere positive polynomial of degree , and . Assuming the existence of such a decomposable polynomial , we built from its coefficients the vector
and prove that is normal to a supporting hyperplane for . Indeed, for any -tuple , the inner product satisfies the following inequality:
Since the ’s are everywhere positive, equality holds if and only if . Given the existence of a decomposable polynomial , this proves that indexes all ’s that lie on a face in , and they of course span by definition of . To prove that is combinatorially equivalent to , it suffices to show that each is in fact a vertex of , since is a projection of . This can be shown with the normal vector , using the same calculation as before.
As in the proof of Proposition 2.5, this ensures that the -skeleton of completely contains the -skeleton of , and we argue that they actually coincide since is furthermore a projection of .
Before showing how to choose the index sets that enable us to construct the polynomials in general, we illustrate the proof on the smallest example. ∎
Let and . Choose the index sets , with , and separated in the sense that the largest element of be smaller than the smallest element of . For any -dimensional face of indexed by , consider the polynomial of degree :
Since the index sets , are separated, the discriminant is negative, which implies that the polynomial is positive for all values of . A symmetric formula holds for the -dimensional faces of whose index sets are of the form .
Proof of Theorem 2.6, continued.
We still need to show how to choose the index sets that enable us to construct the polynomials in general. Once we have chosen these index sets, finding is equivalent to the task of finding polynomials such that
is monic of degree .
The polynomials are equal, up to the coefficients on and .
for all .
The first two items form a linear system equations on the coefficients of the ’s which has the same number of equations as variables, namely . We show that it has a unique solution if one chooses the correct index sets , and we postpone the discussion of requirement (iii) to the end of the proof. To do this, choose distinct reals and look at the similar equation system:
are monic polynomials of degree .
The polynomials are equal, up to the coefficients on and .
The first equation system moves into the second when we deform the points of the sets continuously to , respectively. By continuity of the determinant, if the second equation system has a unique solution then so has the first equation system as long as we chose the sets close enough to the ’s for all . Observe that in the end, we can fulfill all these closeness conditions required for all -faces of since there are only finitely many -faces.
Note that a polynomial of degree has the form
for a monic polynomial and some reals and if and only if has the form
for some polynomial with leading coefficient . The backward direction can be settled by assuming, without loss of generality, that . Indeed, otherwise make a change of variables and then integrate (2) twice (with constants of integration equal to zero) to obtain (1).
Therefore the second equation system is equivalent to the following third one:
are polynomials of degree with leading coefficient .
The polynomials all equal the same polynomial, say .
Since , this system of equations has the unique solution
Therefore, the first two systems of equations both have a unique solution (as long as the ’s are chosen sufficiently close to the ’s). It thus only remains to deal with the positivity requirement (iii).
In the unique solution of the second equation system, the polynomial is obtained by integrating twice with some specific integration constants. For a fixed , we can again assume . Then both integration constants were chosen to be zero for this , hence . Since is non-negative and zero only at isolated points, is strictly convex, hence non-negative and zero only at . Therefore is positive for . Since we chose , we can quickly compute the correspondence between the coefficients of and of :
therefore is everywhere positive. Since the solutions of linear equation systems move continuously when one deforms the entries of the equation system by a homotopy, this ensures that is everywhere positive if is chosen close enough to . The positivity of finishes the proof. ∎
3. Projections of deformed products of simple polytopes
In the previous section, we saw an explicit construction of polytopes whose -skeleton is equivalent to that of a product of simplices. In this section, we provide another construction of -PPSN polytopes, using Sanyal & Ziegler’s technique of “projecting deformed products of polygons” [z-ppp-04, sz-capdd] and generalizing it to products of arbitrary simple polytopes. This generalized technique consists in projecting a suitable polytope that is combinatorially equivalent to a given product of simple polytopes in such a way as to preserve its complete -skeleton. The special case of products of simplices then yields -PPSN polytopes.
3.1. General situation
We first discuss the general setting: given a product of simple polytopes, we construct a polytope that is combinatorially equivalent to and whose -skeleton is preserved under the projection onto the first coordinates.
3.1.1. Deformed products of simple polytopes
Let be simple polytopes of respective dimensions and facet descriptions . Here, each matrix has one row for each of the facets of , and . The product then has dimension , and its facet description is given by the inequalities
The left hand matrix, whose blank entries are all zero, shall be denoted by . It is proved in [az-dpmsp-99] that for any matrix obtained from by arbitrarily changing the zero entries above the diagonal blocks, there exists a right-hand side such that the deformed polytope defined by the inequality system is combinatorially equivalent to . The equivalence is the obvious one: it maps the facet defined by the -th row of to the one given by the -th row of , for all . Following [sz-capdd], we will use this “deformed product” construction in such a way that the projection of to the first coordinates preserves its -skeleton in the following sense.
3.1.2. Preserved faces and the Projection Lemma
For integers , let denote the orthogonal projection to the first coordinates, and denote the dual orthogonal projection to the last coordinates. Let be a full-dimensional simple polytope in , with in its interior. The following notion of preserved faces — see Figure 2 — will be used extensively at the end of this paper:
Definition 3.1 ([z-ppp-04]).
A proper face of a polytope is strictly preserved under if
is a face of ,
and are combinatorially isomorphic, and
The characterization of strictly preserved faces of uses the normal vectors of the facets of . Let denote the facets of . For all , let denote the normal vector to , and let . For any face of , let denote the set of indices of the facets of containing , i.e., such that .
Lemma 3.2 (Projection Lemma [az-dpmsp-99, z-ppp-04]).
A face of the polytope is strictly preserved under the projection if and only if is positively spanning.
3.1.3. A first construction
Let be maximal such that the matrices are entirely contained in the first columns of . Let and . By changing bases appropriately, we can assume that the bottom block of is the identity matrix for each . In order to simplify the exposition, we also assume first that , i.e., that the projection on the first coordinates separates the first block matrices from the last . See Figure 3a.
Let be a set of vectors such that is the Gale transform of a full-dimensional simplicial neighborly polytope — see [z-lp-95, m-ldg-02] for definition and properties of Gale duality. By elementary properties of the Gale transform, has vertices, and . In particular, every subset of vertices spans a face of , so every subset of elements of is positively spanning.
We deform the matrix into the matrix of Figure 3a, using the vectors to deform the top rows. We denote by the corresponding deformed product. We say that a facet of is “good” if the right part of the corresponding row of is covered by a vector of , and “bad” otherwise. Bad facets are hatched in Figure 3a. Observe that there are bad facets in total.
Let be a -face of . Since is a simple -dimensional polytope, is the intersection of facets, among which at least are good facets. If the corresponding elements of are positively spanning, then is strictly preserved under projection onto the first coordinates. Since we have seen that any subset of vectors of is positively spanning, will surely be preserved if , which is equivalent to
Thus, under this assumption, we obtain a -dimensional polytope whose -skeleton is combinatorially equivalent to that of .
3.1.4. When the projection slices a block
We now discuss the case when , for which the method is very similar. We consider vectors such that