CCONDITIONALLY DECOMPOSABLE POLYTOPES
JIE WANG AND DAVID YOST
Abstract.
We construct for the first time conditionally decomposable d -polytopes for d ≥
4. These examples have any number of vertices from 4 d − d − d − d -polytopes that have a linesegment for a summand. Introduction
The existence of conditionally decomposable polytopes has only been well stud-ied in 3-dimensions. In 2-dimensions, no such examples exist: line segments andtriangles are indecomposable and all other polygons are decomposable. In higherdimensions, no examples of conditionally decomposable polytopes had been founduntil now. In this paper, we give explicit examples of conditionally decomposable d -polytopes, for all dimensions d ≥ P is decomposable if it can be written as the Minkowski sumof two polytopes, i.e. P = Q + R = { q + r : q ∈ Q, r ∈ R } , where Q, R are nothomothetic to P . The set of all faces of a polytope is a lattice with respect toinclusion. We say that two polytopes are combinatorially equivalent if there is anisomorphism between their face lattices. It has long been known, at least in threedimensions, that the combinatorial structure of a polytope does not determine itsdecomposability. Thus it is natural to call a polytope combinatorially decomposable if every polytope combinatorially equivalent to it is decomposable, combinatoriallyindecomposable if every polytope combinatorially equivalent to it is indecompos-able, and conditionally decomposable if one polytope combinatorially equivalent toit is decomposable and another one combinatorially equivalent to it is indecompos-able. The concept of decomposability of polytopes was introduced by Gale (with adifferent name); Shephard [10] made perhaps the first serious study of it.The first conditionally decomposable polytope seems to be in the thesis of Meyer[6], the cuboctahedron, which has 12 vertices and 14 faces. If T denotes a tetra-hedron in R , then the difference set T − T = { x − y : x ∈ T, y ∈ T } is easilychecked to be a cuboctahedron. Meyer showed [6, Section 3.1] that it is combinato-rially equivalent to an indecomposable polyhedron. He did this by proving that anypolyhedron combinatorially equivalent to the cuboctahedron is indecomposable, ifone of its quadrilateral faces is not a trapezium, and three of the quadrilateralfaces which touch it are parallelograms, and then exhibiting one example with thisproperty.Kallay [4] gave another example with only 12 faces and 10 vertices, the sum of abi-pyramid over a square and a line segment. By tilting the side faces slightly, he a r X i v : . [ m a t h . C O ] F e b JIE WANG AND DAVID YOST obtained an indecomposable polytope combinatorially equivalent to it. Smilansky[11] made the most detailed study of conditionally decomposable polytopes, estab-lishing the existence of conditionally decomposable 3-polytopes with V vertices and F facets whenever V ≤ F ≤ V −
8, and showing that these are the only valuesopf F and V for which conditionally decomposable 3-polytopes exist. It is theneasy to see that for 3-polytopes, the conditionally decomposable polytopes with 8vertices (and 8 facets) are the ones with the minimum number of vertices. Smi-lansky effectively gave two conditionally decomposable examples with V = F = 8;these turn out to be the examples numbered 282 and 288 in the earlier cataloguedue to Federico [2], of all 3-polytopes with 4 to 8 faces. These two examples wererediscovered in [7] and [14]. We generalize these two examples into d -polytopeswith d ≥
4. There is exactly one other conditionally decomposable example with d = 3 and V = F = 8: the gyrobifastigium, numbered 287 in Federico’s catalogue[14, Proposition 2.10].To continue, we need some sufficent conditions for the indecomposability of poly-topes. Rather than give a detailed account, we just summarize the results which weneed. Kallay [4] defined a geometric graph as any graph whose vertices lie in someEuclidean space, and whose edges are some of the segments between them. Thisincludes the edge graph, i.e. the 1-skeleton, of any polytope. He then generalizedto such graphs the notion of decomposability of polytopes. This generalisation isconsistent, i.e. a polytope P is indecomposable if and only if its 1-skeleton G ( P ) isindecomposable [4, Corollary 5]. By strongly connected triangular chain, we meana sequence of triangles T , · · · , T n such that T i ∩ T i +1 is an edge for each i . By tri-angle in a geometric graph, we mean three vertices, each pair of which determinesan edge of the graph; in the graph of a polytope, this may be, but need not be,a triangular face. For more information on the decomposability of polytopes, werefer to [9] and [3]. The following omnibus result combines work from [10], [4], [7],[8], [14], [13]. Theorem 1.1. (1)
Suppose a geometric graph G contains a strongly connectedtriangular chain, whose union contains every vertex of G . Then G is inde-composable. (2) The graph of any d -polytope with strictly fewer than d vertices contains astrongly connected triangular chain, whose union touches every facet. (3) If the graph of a d -polytope P contains an indecomposable subgraph, whoseunion touches every facet, then P is indecomposable. (4) Let C be the disjoint union of indecomposable geometric graphs A and B ,and suppose there are disjoint edges [ a , b ] , [ a , b ] with a i ∈ A , b i ∈ B , i =1 , . If the lines aff( a , b ) and aff( a , b ) are skew then C is indecomposable. Constructions
Using Kallay’s result, that skew lines connecting two indecomposable subgraphsmakes the union of the subgraphs indecomposable, we present our first construction.
Example 2.1.
For each d ≥ , there is a conditionally decomposable d -polytopewith d − vertices.Proof. Let P be the polytope whose 4 d − ONDITIONALLY DECOMPOSABLE POLYTOPES 3 A i = e i , for i = 1 , . . . , d − , where e i is the usual unit vector in R d ,A d − = (0 , . . . , , the origin ,B i = A i + (0 , . . . , , , , , for i = 1 , . . . , d − ,C i = A i + (0 , . . . , , , , , for i = 1 , . . . , d − ,D i = A i + (0 , . . . , , , , , for i = 1 , . . . , d − ,A = (0 , . . . , , , , ,B = (0 , . . . , , , , ,C = (0 , . . . , , , , ,D = (0 , . . . , , , , . Then P is a d -polytope that has a line segment for a summand (in the directionof e d − ), so P is decomposable. Since two polytopes are combinatorially equivalentif and only if there is a one-one correspondence between their vertices and facets,which preserves the inclusion relation, we first list the following facets:conv { A i , B i , C i , D i : 1 ≤ i ≤ d − } with supporting hyperplane x d = 0 , conv { A i , B i , C i , D i , A, B, C, D : 1 ≤ i ≤ d − , i (cid:54) = j } for each j ∈ { , . . . , d − } , with supporting hyperplane x j = 0 , conv { A i , B i , C i , D i , A, B, C, D : 1 ≤ i ≤ d − } with supporting hyperplane x + · · · + x d − + x d = 1 , conv { A i , C i , A, C : 1 ≤ i ≤ d − } with supporting hyperplane x d − = 0 , conv { B i , D i , B, D : 1 ≤ i ≤ d − } with supporting hyperplane x d − = 1 , conv { C i , C, D : 1 ≤ i ≤ d − } with supporting hyperplane − x d − + x d − + x d = 3 , conv { A i , A, B : 1 ≤ i ≤ d − } with supporting hyperplane x d − + x d − − x d = 0 , conv { C i , D i , D : 1 ≤ i ≤ d − } with supporting hyperplane x d − = 3 , conv { A i , B i , B : 1 ≤ i ≤ d − } with supporting hyperplane x d − = 0 . To show that there are no other facets, we first denote the first listed facet by ∆,and notice that it has the form ∆(1 , , d − d − P is the convex hull of this facet and four additionalvertices A, B, C, D . Note also that
A, C are not adjacent to B i , D i , and B is notadjacent to C i , and D is not adjacent to A i , since those line segments are notexposed.Let H denote the hyperplane { x ∈ R d : x d − = } . Then P ∩ H is a cross sectionthat separates the vertices of P into two parts { A, A i , B, B i } and { C, C i , D, D i } ,and is in fact a ( d − d − H ) of P is d + 1, becausethere is a one-one correspondence between facets of P ∩ H and the side facets of P . To show that there are no other upper and lower facets (i.e. facets contained inone of the half spaces whose boundary is H ), we use the fact that a facet of P mustbe of the form conv { F ∪ F } , where F is a face of ∆, and F ⊆ { A, B, C, D } . Interms of its upper facets, the only possible faces of ∆(1 , , d −
3) to constitute partof a facet of P are its ( d − { C i , D i : 1 ≤ i ≤ d − } , and ( d − { C i : 1 ≤ i ≤ d − } , conv { D i : 1 ≤ i ≤ d − } , conv { C i , D i : except one index } .Since a facet of P has dimension d −
1, and C is not adjacent to D i , the only possibleupper facets are those shown before. Similarly, we can show that the only possiblelower facets are those shown before. JIE WANG AND DAVID YOST
To obtain P (cid:48) , we perturb C and C i slightly. Replace C i with C (cid:48) i = A i +(0 , . . . , , εε − , , i = 1 , . . . , d −
2, replace C with C (cid:48) = (0 , . . . , , − ε, − ε, ε > P (cid:48) of facets of P are still facets, and again we list thefacets and calculate their supporting hyperplanes:conv { A i , B i , C (cid:48) i , D i : 1 ≤ i ≤ d − } with supporting hyperplane x d = 0 , conv { A i , B i , C (cid:48) i , D i , A, B, C (cid:48) , D : 1 ≤ i ≤ d − , i (cid:54) = j } for each j ∈ { , . . . , d − } , with supporting hyperplane x j = 0 , conv { A i , B i , C (cid:48) i , D i , A, B, C (cid:48) , D : 1 ≤ i ≤ d − } , with supporting hyperplane x + · · · + x d − + x d = 1 , conv { A i , C (cid:48) i , A, C (cid:48) : 1 ≤ i ≤ d − } , with supporting hyperplane ε − ε x d − − x d − + x d = 0 , conv { B i , D i , B, D : 1 ≤ i ≤ d − } with supporting hyperplane x d − = 1 , conv { C (cid:48) i , C (cid:48) , D : 1 ≤ i ≤ d − } , with supporting hyperplane ( ε − x d − + (1 − ε ) x d − + (2 ε + 1) x d = 3 , conv { A i , A, B : 1 ≤ i ≤ d − } with supporting hyperplane x d − + x d − − x d = 0 , conv { C (cid:48) i , D i , D : 1 ≤ i ≤ d − } with supporting hyperplane x d − = 3 , conv { A i , B i , B : 1 ≤ i ≤ d − } with supporting hyperplane x d − = 0 . Among the facets of P , ∆ corresponds in P (cid:48) to ∆ (cid:48) , which is the Minkowski sum ofconv { A i , B i : 1 ≤ i ≤ d − } and conv { A d − , C (cid:48) d − } , and it is not difficult to see thatit is also combinatorially equivalent to ∆. So, to know the 1-skeleton of P (cid:48) , we justneed to know the adjacency relations between { A, B, C (cid:48) , D } and ∆ (cid:48) . Now C (cid:48) and B i are not adjacent because B i + C (cid:48) = ε B i + A + (1 − ε ) ε ) D i + ε (1 − ε )2(1+2 ε ) C (cid:48) i , C (cid:48) and D i are not adjacent because D i + C (cid:48) = ε A i + − ε ε ) C (cid:48) i + ε ε ) D i + D , B and C (cid:48) i are not adjacent because B + C (cid:48) i = − ε − ε ) A i + ε − ε ) C (cid:48) i + − ε − ε − ε ) D + ε − ε ) D i , as those edges are not exposed. Since P ⊂ P (cid:48) , and the line segments AB i , AD i , DA i are not exposed in P , they are also not exposed in P (cid:48) . Also A i , C (cid:48) are still not adjacent since all A i C (cid:48) i are parallel to AC (cid:48) , meaning that for each i ,conv { A i , C (cid:48) i , A, C (cid:48) } is still a planar quadrilateral.Taking a cross section P (cid:48) ∩ H , one can show that it is a simple ( d − d − P (cid:48) ∩ H iscombinatorially equivalent to a prism based on a ( d − P (cid:48) . And finally, we conclude that P (cid:48) is combinatoriallyequivalent to P .Notice that conv { C (cid:48) , C (cid:48) i , D : 1 ≤ i ≤ d − } is a ( d − P (cid:48) , and conv { D, D i : 1 ≤ i ≤ d − } is a ( d − { D, D i , B, B i : 1 ≤ i ≤ d − } that is a facet of P (cid:48) , so those corresponding edgesare still edges of P (cid:48) . Therefore, the upper subgraph of P (cid:48) is indecomposable sinceit has a strongly connected triangular chain, and likewise so is the lower subgraphof P (cid:48) . As the line containing A d − C (cid:48) d − (and in fact, every index) and the linecontaining BD are skew, by Theorem 1.1, P (cid:48) is an indecomposable polytope. Weconclude that P is a conditionally decomposable polytope. (cid:3) ONDITIONALLY DECOMPOSABLE POLYTOPES 5
Remark 2.2.
Gluing a proper d -simplex over one of the ( d − -simplex facets of P and correspondingly over one of the ( d − -simplex facets of P (cid:48) gives the existenceof conditionally decomposable polytope with one more vertex (and d − more facets),see [8, Proposition 6] . After gluing a proper d -simplex over a ( d − -simplex facet,these new polytopes still have facets that are ( d − -simplexes, without changingits decomposability, we follow this process indefinitely, which gives the existence ofconditionally decomposable polytopes with more than d − vertices, of which oneof the -faces is the graph numbered 282 of Federico’s list [2] . Remark 2.3 (Another example) . Taking the same base facet ∆ = conv { A i , B i , C i , D i :1 ≤ i ≤ d − } as before, we now add four different vertices ¯ A = (0 , . . . , , , − , , ¯ B = (0 , . . . , , , , , ¯ C = (0 , . . . , , , , , ¯ D = (0 , . . . , , , , . Then one can show that ¯ P = conv { ∆ , ¯ A, ¯ B, ¯ C, ¯ D } is a d -polytope with d − vertices, d + 5 facets, and ¯ P is the sum of a line segment and a -fold pyramid.To generate another polytope that is combinatorially equivalent to ¯ P but in-decomposable, we perturb ¯ C with ¯ C (cid:48) = (0 , . . . , , − ε, − ε, , perturb ¯ C i with ¯ C (cid:48) i = A i + (0 , . . . , , εε − , , , i = 1 , . . . , d − , where ε > is pretty small, andretain the rest of the vertices of ¯ P , then we obtain ¯ P (cid:48) . By the very similar approach,one can show that ¯ P (cid:48) is combinatorially equivalent to ¯ P but indecomposable. Bygluing a proper d -simplex over one simplex facets of ¯ P and ¯ P (cid:48) , respectively, weobtain another construction of conditionally decomposable polytope with d − ormore vertices, of which one of the -faces is the graph numbered 288 of Federico’slist. Remark 2.4.
It is also possible to construct a conditionally decomposable d -polytopethat inherits directly from Kallay’s example with d − vertices. Start with a d -dimensional ∆(1 , , d − , say Q , with vertices A i = e i , i = 1 , . . . , d − ,A d − = (0 , . . . , , the origin ,B i = A i + (0 , . . . , , , , i = 1 , . . . , d − ,C i = A i + (0 , . . . , , , , i = 1 , . . . , d − ,D i = A i + (0 , . . . , , , , i = 1 , . . . , d − . This time we perturb Q first, replace C i with C (cid:48) i = A i + (0 , . . . , , − ε, , i =1 , . . . , d − , retain the rest of vertices and we obtain Q (cid:48) that is combinatoriallyequivalent to Q but with skew lines. Stacking an appropriate pyramid over facets of Q , conv { C i , D i : i = 1 , . . . , d − } and conv { A i , B i : i = 1 , . . . , d − } , respectivelyand also an appropriate pyramid over facets of Q (cid:48) , conv { C (cid:48) i , D i : i = 1 , . . . , d − } and conv { A i , B i : i = 1 , . . . , d − } , respectively. Since the geometric graph of apyramid is indecomposable, we obtain one polytope decomposable and another oneindecomposable but with the same face lattice, by [8, Proposition 6] . This also showsanother construction of conditionally decomposable polytopes with more than d − vertices. Figure 1 gives the sketch of those examples when d = 4. JIE WANG AND DAVID YOST (1) P (2) ¯ P (3) 4 d − Figure 1. d -polytope with no more than 2 d − Lemma 2.5.
Let P be a d -polytope, [0 , a ] be a line segment. Then P = P + [0 , a ] and P = P + [0 , ka ] are combinatorially equivalent, k > .Proof. We first show that there is a one-one correspondence between V ( P ) and V ( P ). Let v ∈ V ( P ). Then1) v ∈ V ( Q ). Then there exists a hyperplane H v = { x : α · x = c } such that α · v = c,α · ( v + a ) > c, which implies α · q > c,α · ( q + a ) > c, α · v = c,α · ( v + ka ) = α · v + kα · a = c + kα · a > c,α · q > c,α · ( q + ka ) = α · q + kα · a > c + α · a > c, where q ∈ V ( Q ), q (cid:54) = v . So H v is also a supporting hyperplane of P at v , i.e. v isalso a vertex of P .2) v ∈ V ( Q ) + a . Then there exist q v ∈ V ( Q ) such that v = q v + a . Similarly,there exists a hyperplane H v = { x : α · x = c } such that α · ( q v + a ) = c,α · q v > c, which implies α · q > c,α · ( q + a ) > c, α · ( q v + ka ) = c + ( k − α · a,α · q v = c − α · a > c + ( k − α · a,α · q > c − α · a > c + ( k − α · a,α · ( q + ka ) > c + ( k − α · a, where q ∈ V ( Q ), q (cid:54) = q v . This suggests H (cid:48) v = { x : α · ( q v + ka ) = c + ( k − α · a } is a supporting hyperplane of P at v (cid:48) = q v + ka , a vertex of P .Conversely, let v ∈ V ( P ). Then v ∈ V ( Q ) or v ∈ V ( Q ) + ka . By the sameapproach, one can show that for every vertex of P , there is a corresponding vertexof P . So there is a one-one correspondence between V ( P ) and V ( P ).Let F be a facet of P , then F = F + F , where F is a face of Q , and F is a faceof [0 , a ]. Denote Q = conv { q i : i ∈ I } . If F is the origin, then F = conv { q i : i ∈ I } for some index set I ⊂ I such that ONDITIONALLY DECOMPOSABLE POLYTOPES 7 α · q i = c, i ∈ I ,α · q i > c, i ∈ I \ I , which implies α · ( q i + a ) > c, i ∈ I. α · q i = c, i ∈ I ,α · q i > c, i ∈ I \ I ,α · ( q i + ka ) = α · q i + kα · a > c, i ∈ I. So, F (cid:48) = F is also a facet of P . Similarly, one can show that if F = a , F =conv { q i + a : i ∈ I ⊂ I } , then F (cid:48) = conv { q i + ka : i ∈ I } is a facet of P , and if F = [0 , a ], F = conv { q i , q i + a : i ∈ I ⊂ I } , then F (cid:48) = conv { q i , q i + ka : i ∈ I } isa facet of P . One can also show that for a facet of the form conv { q j : j ∈ J ⊂ I } of P , conv { q j : j ∈ J } is a facet of P , for a facet of the form conv { q j + ka : j ∈ J ⊂ I } of P , conv { q j + a : j ∈ J } is a facet of P , and for a facet of the formconv { q j , q j + ka : j ∈ J ⊂ I } of P , conv { q j , q j + a : j ∈ J } is a facet of P .Therefore, there is a one-one correspondence between vertices and facets of P and P , of which the inclusion holds, so, P and P are combinatorially equivalent. (cid:3) The main theorem then follows rather straightforwardly.
Theorem 2.6.
Let P be a d -polytope with no more than d − vertices that has aline segment for a summand. Then P is combinatorially decomposable.Proof. Since the decomposability of a polytope is translation and rotation invariant,we can assume that P has [0 , a ] for a summand. By Lemma 2.5 we can partitionvertices of P into two sets A and B such that for each vertex in A , there is at mostone vertex in B that is adjacent to it. Take two parallel hyperplanes H , H suchthat they both separate A and B . Let C = P ∩ H , C = P ∩ H . Then C and C are ( d − C (respectively C ) as { v i : i ∈ I } (respectively { v (cid:48) i : i ∈ I } ) in such a way that[ v i , v (cid:48) i ] is parallel to [0 , a ], for each i ∈ I .Let F = conv { v i : i ∈ I ⊂ I } be a facet of C , then there is a unique facet F of P such that F = F ∩ H . Then F := F ∩ H = conv { v (cid:48) i : i ∈ I } is a correspondingfacet of C , and vice versa. It follows that C and C are combinatorially equivalent.Since at least one of A, B has no more than 2 d − d − C and C . So there exists a strongly connected triangularchain in both C and C , and that the corresponding pair of each of these trianglesconv { v i , v i , v i } , conv { v (cid:48) i , v (cid:48) i , v (cid:48) i } , is part of a 3-face of P . So, the line l i con-taining v i , v (cid:48) i , the line l i containing v i , v (cid:48) i and the line l i containing v i , v (cid:48) i arepairwise coplanar. And this is also the case for any polytope that is combinatoriallyequivalent to P , since those lines contains a 1-face. Also, in the strongly connectedtriangular chain, two consecutive triangles share a common edge, so correspond-ingly two consecutive 3-faces of P (that cross C and C ) have two common lineswhich are coplanar. Note that l i , l i , l i are either all parallel or concurrent. Soall these lines are either all parallel or concurrent, where the concurrent point doesnot belong to P . In the latter case, by taking a suitable projective transformationthrough the concurrent point without intersecting with P , we obtain a d -polytopecombinatorially equivalent to P with all such those edges are parallel. Since thedecomposability of a polytope is projective invariant [5], it then suffices to see thatby defining one side of the vertices along the direction of those parallel edges givesthe combinatorial decomposability of P . (cid:3) However, whether there is a higher dimensional example of which one of the3-faces is the graph numbered 287 of Federico’s list is still a mystery.Deciding the existence of higher dimensional conditionally decomposable poly-topes with triangles for summands may be the next line of investigation.
JIE WANG AND DAVID YOST
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