A lower bound theorem for d-polytopes with 2d+1 vertices
AA LOWER BOUND THEOREM FOR d -POLYTOPES WITH D + 1 VERTICES
GUILLERMO PINEDA-VILLAVICENCIO
Centre for Informatics and Applied Optimisation, Federation University, AustraliaSchool of Information Technology, Deakin University, Geelong, Australia
DAVID YOST
Centre for Informatics and Applied Optimisation, Federation University, Australia
Abstract.
We establish the exact lower bound for the number of k -facesof d -polytopes with 2 d + 1 vertices, for each value of k , and characterise theminimisers. As a byproduct, we characterise all d -polytopes with d +3 vertices,and only one or two edges more than the minimum. Introduction
A polytope P of dimension d is a d -polytope and a k -dimensional face of thepolytope is a k -face. A facet is a ( d − ridge is a ( d − edge is a1-face, and a vertex is a 0-face. The number of k -faces of P is denoted by f k ( P ).The graph of a polytope is the graph formed by the vertices and edges of thepolytope. For an edge e = xy of any graph, we say that x and y are adjacent or neighbours ; that x and y are incident with e ; and that e is incident with x and y .(We denote edges of graphs and polytopes by concatenation of their names, ratherthan using the interval notation.)Lower bound theorems for the number of faces of polytopes have attracted theattention of numerous researchers over the years [2, 11–13, 19]. The lower boundtheorem for simple polytopes of Barnette [1, 2] is the best known such theorem;a vertex in a d -polytope is simple if it is incident with exactly d edges; otherwiseit is nonsimple . A polytope is simple if all its vertices are simple; otherwise it is nonsimple . E-mail addresses : [email protected], [email protected] . Date : February 26, 2021.2020
Mathematics Subject Classification.
Primary 52B05; Secondary 52B12.
Key words and phrases. polytope, f-vector, dual polytope, excess degree. a r X i v : . [ m a t h . C O ] F e b A LOWER BOUND THEOREM FOR D -POLYTOPES WITH 2 D + 1 VERTICES Theorem 1 (Simple polytopes, [1, 2]) . Let d (cid:62) and let P be a simple d -polytopewith f d − facets. Then (1) f k ( P ) (cid:62) ( d − f d − − ( d + 1)( d − , if k = 0; (cid:0) dk +1 (cid:1) f d − − (cid:0) d +1 k +1 (cid:1) ( d − − k ) , if k ∈ [1 , d − . Moreover, for every value of f d − (cid:62) d + 1 , there are simple d -polytopes for whichequality holds. Dually, Barnette’s Theorem also gives the exact lower bound for the numbers offaces of each positive dimension of a simplicial polytope, in terms of the number ofvertices. We are interested in finding optimal lower bounds for the number of k -faces of a general polytope, with a given number of vertices. We break this problemdown according to the number of vertices.Gr¨unbaum [10, Sec. 10.2] defined a function φ k ( d + s, d ), for s (cid:54) d , by:(2) φ k ( d + s, d ) = (cid:18) d + 1 k + 1 (cid:19) + (cid:18) dk + 1 (cid:19) − (cid:18) d + 1 − sk + 1 (cid:19) . That φ k ( d + s, d ) gives the minimum number of k -faces of a d -polytope with d + s vertices was conjectured by Gr¨unbaum [10, Sec. 10.2] and recently proved by Xue[19], who also characterised the unique minimisers for k ∈ [1 , d − k in [17].) The minimisers are called triplicesand were defined in [17, Sec. 3]. The ( s , d − s )-triplex M ( s, d − s ) is a ( d − s )-foldpyramid over a simplicial s -prism for s ∈ [1 , d ]. In particular, M (1 , d −
1) is a d -simplex T ( d ) and M ( d,
0) is a simplicial d -prism. The prism over a polytope Q is the product of Q and a line segment, or any polytope combinatorially equivalentto it. By a simplicial d -prism we mean a prism over a simplex. Theorem 2 ( d -polytopes with at most 2 d vertices, [19]) . Let d (cid:62) and let P be a d -polytope with d + s vertices, where s (cid:54) d . Then f k ( P ) (cid:62) φ k ( d + s, d ) for all k ∈ [1 , d − . Also, if f k ( P ) = φ k ( d + s, d ) for some k ∈ [1 , d − then P is the ( s, d − s ) -triplex. This paper extends this result to the case of d -polytopes with 2 d + 1 vertices.That is, we establish the minimum number ψ k ( d ) of k -faces of a d -polytope with2 d + 1 vertices, and characterise the minimisers (Theorem 3).Note that the case k = d − M ( s, d − s ) has exactly φ d − ( d + s, d ) = d + 2facets, but it may not be the only minimiser. For example, a pyramid over ∆(2 , k = d − d -polytopes with even more than 2 d vertices. He showed that theminimal number of facets, for polytopes satisfying 2 d (cid:54) f (cid:54) d + 2 d , is either d + 2 or d + 3, depending on number theoretic properties of d and f . For the case LOWER BOUND THEOREM FOR d -POLYTOPES WITH 2 D + 1 VERTICES 3 of 2 d + 1 vertices, we have ψ d − ( d ) = d + 3 when d is prime, while ψ d − ( d ) = d + 2when d is composite. Our main result Theorem 3 shows that this sort of dichotomyis quite common.The case k = 1 of Theorem 3 was settled in [17]; we restate this result hereas Theorem 11. The case k = d − §
3. Building on this, wedescribe in § k . With one exception in dimension three,each minimiser is either a d -pentasm or a certain d -polytope with d + 2 facets. Letus describe these examples briefly. More details appear in Section 2.5 below.The pentasm P m ( d ) in dimension d , or d -pentasm , was defined in [17, Section 4];it can be obtained by truncating a simple vertex from the triplex M (2 , d − (Minkowski) sum of two polytopes Q + R is defined to be { x + y : x ∈ Q, y ∈ R } . We define the simple polytope ∆( r, s ) (with r, s >
0) as the sum ofan r -dimensional simplex and an s -dimensional simplex, lying in complementarysubspaces; it has r + s + 2 facets. It turns out that t -fold pyramids over suchsimple polytopes, denoted ∆ t ( r, s ), are the only d -polytopes with d + 2 facets; seeLemma 12 below. This can be established by dualising the structure of d -polytopeswith d + 2 vertices [10, Sec. 6.1], and was first done explicitly in [13].Finally, we introduce the other minimiser in dimension three. In R d , denote byΣ( d ) a polytope that is combinatorially equivalent to the convex hull of { , e , e + e k , e , e + e k , e + e , e + e + 2 e k : 3 (cid:54) k (cid:54) d } , where e i is the standard i th unit vector in R d . This polytope has 3 d − d + 3 facets. So for the time being, it is mainly of interest to us when d = 3.Examples of all the aforementioned polytopes are depicted in Fig. 1. The nexttheorem summarises our results; the f -vector of a d -polytope P , denoted f ( P ), isthe sequence ( f ( P ) , . . . , f d − ( P )) of the numbers of faces of P of different dimen-sions. Theorem 3.
Let d (cid:62) and consider the class of d -polytopes with d + 1 vertices.Fix k ∈ [1 , d − . Then the following hold. (i) Let d = 3 . If P is Σ(3) or a pentasm, then f ( P ) = (7 , , . Otherwise, f ( P ) > and f ( P ) > . (ii) Let d = 4 . If P is ∆(2 , , then f ( P ) = (9 , , , . If P is a pentasm, then f ( P ) = (9 , , , . Otherwise, f ( P ) > , f ( P ) > , and f ( P ) (cid:62) . (iii) Let d = 5 . If P is a pentasm, then f ( P ) = (11 , , , , . Otherwise, f ( P ) > , f ( P ) > , f ( P ) > , and f ( P ) (cid:62) . (iv) If d (cid:62) and d is prime, then the d -pentasm is the unique minimiser of f k ( P ) . A LOWER BOUND THEOREM FOR D -POLYTOPES WITH 2 D + 1 VERTICES (d)(a) (b) (c) v v v v v v v v u v u u u u u Figure 1.
Schlegel diagrams of extremal polytopes with 2 d + 1vertices. (a) Σ(3). (b) 3-pentasm. (c) 4-pentasm. (d) ∆(2 , If d (cid:62) and d is composite, then the minimiser of f k ( P ) is either a d -pentasm or a ( d − r − s ) -fold pyramid over ∆( r, s ) where d = rs is anontrivial factorisation of d . Preliminaries
Unless otherwise stated the notation and terminology follows from [20]. In par-ticular, we assume familiarity with duality of polytopes, [10, § § § Truncation of polytopes.
Recall that a vertex in a d -polytope P is simpleif and only if it is contained in exactly d facets. A nonsimple vertex in P may besimple in a proper face of P ; we often need to make this distinction. Indeed, everyvertex is simple in every 2-face which contains it.A fundamental tool for the construction of new polytopes is the truncation ofa face [4, p. 76]. An extension of this idea is the truncation of a set of verticeswhich do not necessarily form a face. This idea is implicit in [10, § H be a hyperplane intersecting the interior of P and containing no vertex of P , andlet H + and H − be the two closed half-spaces bounded by H . Set P (cid:48) := H + ∩ P .Denoting by X the set of vertices of P lying in H − , the polytope P (cid:48) is said to beobtained by truncating the set X by H . One often says that P has been sliced or cut at X . We do not assume that X forms the vertex set of any face of P . Proposition 4.
Let P be a d -polytope with vertex set V and let X ⊂ V be acollection of vertices for which the convex hulls of X and V \ X are disjoint. Supposethat H is a hyperplane strictly separating X and V \ X , and that H + and H − arethe two closed half-spaces bounded by H .Let P (cid:48) := H + ∩ P be obtained by truncating X by H . In addition suppose that,whenever vw is an edge of P with v ∈ H + and w ∈ H − , at least one of the vertices v, w is simple in P .Then every vertex in the facet H ∩ P is simple in P (cid:48) , and thus the facet H ∩ P is a simple ( d − -polytope. LOWER BOUND THEOREM FOR d -POLYTOPES WITH 2 D + 1 VERTICES 5 Proof.
Any vertex u in the facet H ∩ P is the intersection of H and an edge e of P with one endpoint in H + and the other in H − . Since this endpoint is containedin only d facets of P , the edge e itself is contained in exactly d − P .Consequently, the facets of P (cid:48) containing the vertex u are precisely the facets of P containing the edge e plus the facet H ∩ P . Thus the vertex u in P (cid:48) is containedin exactly d facets of P (cid:48) , and is simple in P (cid:48) . (cid:3) With the notation of Proposition 4, if X = { v } then the facet P ∩ H of P (cid:48) is called the vertex figure P/v of P at v . Then there is a bijection between the k -faces of P that contains v , and the ( k − P/v , [14, Theorem 16] or [20,Prop. 2.4]. Some routine corollaries of Proposition 4 read as follows.
Lemma 5.
Let P be a d -polytope. (i) Let v be a vertex whose neighbours are all simple in P . Then the vertexfigure of P at v is a simple ( d − -polytope. (ii) Let v be a simple vertex in P , and P (cid:48) the polytope obtained by truncating v . Then f ( P (cid:48) ) = f ( P ) + d − and f k ( P (cid:48) ) = f k ( P ) + (cid:0) dk +1 (cid:1) for k (cid:62) . (iii) Let v be a simple vertex in P . Then v and every k of its d neighbours definea k -face of the polytope P , and every k -face containing v is defined in thisway. (iv) Let e be an edge in P whose endpoints are both simple vertices, and P (cid:48) the polytope obtained by truncating e . Then f ( P (cid:48) ) = f ( P ) + 2 d − and f ( P (cid:48) ) = f ( P ) + d − d . (v) If some 2-face of a simple 4-polytope P has m + 1 or more vertices, thenthere is another simple 4-polytope P (cid:48) with f ( P (cid:48) ) = f ( P ) + m + 2 and f ( P (cid:48) ) = f ( P ) + 1 . Proof.
The proofs of (i) and (ii) are well-known direct consequences of Proposi-tion 4. Part (iii) follows from the vertex figure
P/v of P at v being a ( d − P ∩ H of P being a ( d − X := { v , . . . , v m } be a set of m vertices in this 2-face with v i adjacentto v i +1 for each meaningful i . The convex hull of X is clearly disjoint from theconvex hull of the other vertices Y of P , and so there is a hyperplane H separatingthese two sets. We let H + be the closed halfspace determined by H that contains Y and let P (cid:48) be the polytope P ∩ H + .Since v and v m both have three neighbours in Y , while v , . . . , v m − each havetwo neighbours in Y , there are altogether 3 × m −
2) edges between X and Y , whence there are 3 × m −
2) vertices in P (cid:48) that are not in P . In addition,the m vertices of X are not in P (cid:48) , so f ( P (cid:48) ) = f ( P ) + m + 2. It is plain that f ( P (cid:48) ) = f ( P ) + 1. (cid:3) In particular, truncation of a simple polytope gives us another simple polytopewith more vertices. This is a useful technique for producing examples of simple
A LOWER BOUND THEOREM FOR D -POLYTOPES WITH 2 D + 1 VERTICES polytopes. However not all simple polytopes can be produced this way, as truncat-ing a simple vertex in a polytope results in a polytope with a simplex facet; thecube and the dodecahedron are simple, but have no such facet. More importantexamples for us are the duals of cyclic polytopes, discussed in Section 2.7 below.2.2. Binomial identities.
We require three elementary identities, which can befound in numerous sources, e.g. [9, p. 174].( a − b ) (cid:18) ab (cid:19) = a (cid:18) a − b (cid:19) (3) (cid:18) ab (cid:19) = (cid:18) a − b − (cid:19) + (cid:18) a − b (cid:19) (4) d (cid:88) j =0 (cid:18) jk (cid:19) = (cid:18) d + 1 k + 1 (cid:19) (5)2.3. Excess degree.
The degree of a vertex u in a polytope P , denoted deg P ( u ),is the number of edges in P that are incident with u ; we write deg F ( u ) to denotethe degree of u in the face F of P .The excess degree ξ , or just excess , of a vertex u in a d -polytope is defined asdeg P ( u ) − d , and the excess degree ξ ( P ), or simply excess , of a d -polytope P is thesum of the excess degrees of its vertices; we denote by V ( P ) the set of vertices of P . That is, we have that(6) ξ ( P ) = (cid:88) u ∈ V ( P ) (deg u − d ) = 2 f − df . Remark . If the dimension of a polytope is even, then so is its excess.The excess theorem next proves to be helpful when understanding the structureof polytopes. The paper [16] provides further structural results of d -polytopes withexcess d − Theorem 7 (Excess, [16, Thm. 3.3]) . Let d (cid:62) and let P be a d -polytope. Thenthe smallest possible values of its excess are 0 and d − . Lemma 8 (Excess d −
2, [16, Thm. 4.10]) . Let d (cid:62) and let P be a d -polytopewith excess exactly d − . Then P either (i) has a unique nonsimple vertex, which is the intersection of two facets, or (ii) has d − vertices of excess degree one, which form a ( d − -simplex whichis the intersection of two facets.In either case, the two intersecting facets are both simple polytopes. More on simplicial prisms.
A simplicial d -prism has d + 2 facets: two( d − d simplicial ( d − Remark . Any two of the edges of a simplicial d -prism that join a vertex in onesimplex facet and a vertex in the other simplex facet determine a 2-face, and hence LOWER BOUND THEOREM FOR d -POLYTOPES WITH 2 D + 1 VERTICES 7 are coplanar. It follows that all d such edges are either parallel or contained inconcurrent lines.2.5. More on pentasms and polytopes with d + 2 facets. The definition oftruncation together with (2) easily yields the number of k -faces of a pentasm.(7) f k ( P m ( d )) = (cid:0) d +1 k +1 (cid:1) + (cid:0) dk +1 (cid:1) + (cid:0) d − k (cid:1) , if k (cid:62) d + 1 , if k = 0 . In one concrete realisation, the d -pentasm is the convex hull of the vectors 0, e i for i ∈ [1 , d ], and e + e + e i for i ∈ [1 , d ]. The d -pentasm has d + 3 simple verticesand d − Remark . Label the vertices of any pentasm as u , . . . , u d , v , v , . . . , v d , in sucha way that the edges are u i v i for 1 (cid:54) i (cid:54) d , u i u j for 1 (cid:54) i < j (cid:54) d , and v i v j for 0 (cid:54) i < j (cid:54) d except when ( i, j ) = (1 ,
2) (Fig. 1(b)-(c)). The d -dimensionalpentasm has precisely d + 3 facets:(i) d − i ∈ [3 , d ], the face generatedby all vertices except u i and v i is a pentasm facet),(ii) two simplicial prisms (one generated by all vertices except u , v , v andthe other generated by all vertices except u , v , v ),(iii) and three simplices (one generated by all u i , another generated by all v i except v , and the third generated by all v i except v .In general, pentasms have the minimum number of edges of d -polytopes with2 d + 1 vertices. Theorem 11 (2 d + 1 vertices, [17]) . The d -polytopes with d + 1 vertices and d + d − or fewer edges are as follows. (i) For d = 3 , there are exactly two polyhedra with 7 vertices and 11 edges; P m (3) and
Σ(3) . None have fewer edges. (ii)
For d = 4 , a sum of two triangles ∆(2 , is the unique polytope with 18edges, and the P m (4) is the unique polytope with 19 edges. None have feweredges. (iii)
For d (cid:62) , P m ( d ) is the unique d -polytope with d + d − edges, or equiv-alently, with excess degree d − . None have fewer edges. McMullen [13, Sec. 3] provided expressions for the number of k -faces of a d -polytope with d + 2 facets. Lemma 12 ([13, Sec. 3]) . Let P be a d -dimensional polytope with d + 2 facets,where d (cid:62) . Then, there exist r > , s > and t (cid:62) such that d = r + s + t , and P is a t -fold pyramid over ∆( r, s ) . The number of k -dimensional faces of P is (8) f k ( P ) = (cid:18) r + s + t + 2 k + 2 (cid:19) − (cid:18) s + t + 1 k + 2 (cid:19) − (cid:18) r + t + 1 k + 2 (cid:19) + (cid:18) t + 1 k + 2 (cid:19) . A LOWER BOUND THEOREM FOR D -POLYTOPES WITH 2 D + 1 VERTICES We list the d + 2 facets of such polytopes. Remark . The r + s + t + 2 facets of a t -fold pyramid over ∆( r, s ) are as follows:(i) r + 1 facets that are t -fold pyramids over ∆( r − , s ),(ii) s + 1 facets that are t -fold pyramids over ∆( r, s − t facets that are ( t − r, s ).McMullen actually characterised the values of f for which there is a d -polytopewith d + 2 facets and f vertices; we state a special case. Corollary 14 ([13, special case of Thm. 2]) . If d is prime then there is no d -polytopewith d + 2 facets and d + 1 vertices. Capped prisms.
We define another family of polytopes related to the pen-tasms. For 3 (cid:54) (cid:96) (cid:54) d , let Q (cid:96) be a bipyramid over an ( (cid:96) − P (cid:96),d bea ( d − (cid:96) )-fold pyramid over Q (cid:96) . Then P (cid:96),d is a d -polytope with d + 2 vertices, onlytwo of which are simple, say v and v . Truncating one simple vertex from P (cid:96),d ,say v , gives us a capped d -prism , which we denote by CP ( (cid:96), d ). This definitionis consistent with the one given in [16, Remark 2.11]. All of the d − d + 1vertices and d + d edges, one more than the pentasm. Nevertheless their combina-torial type depends on the value of (cid:96) ; CP ( (cid:96), d ) has d + (cid:96) +1 facets, so their f -vectorsare all distinct. (For (cid:96) = 2, the same construction leads to the d -pentasm, with d + d − v remains a vertex of CP ( (cid:96), d ). It is also the casethat the convex hull of the other 2 d vertices in CP ( (cid:96), d ) is a simplicial d -prism,and v is beyond one simplex facet of this prism, and not beyond any of the otherfacets; hence the name capped prism. A point is said to be beyond a facet [10, § Lemma 15. If (cid:54) (cid:96) (cid:54) d , then, for each (cid:54) k (cid:54) d − , we have that f k ( CP ( (cid:96), d )) > f k ( P m ( d )) . Proof. A d -pentasm is obtained by truncating a simplex vertex from the triplex M (2 , d −
2) and a capped prism is obtained by truncating a simplex vertex fromthe aforementioned P (cid:96),d . Since P (cid:96),d has d +2 vertices, it has strictly more faces of allnonzero dimensions than M (2 , d − (cid:3) Duals of cyclic polytopes.
The moment curve in R d is defined by x ( t ) :=( t, t , . . . , t d ) for t ∈ R , and the convex hull of any n > d distinct points on it givesa cyclic polytope C ( n, d ). These were first defined in [8]. The combinatorial typeof the polytope does not depend on which points on the moment curve have beenchosen; see e.g. [10, § LOWER BOUND THEOREM FOR d -POLYTOPES WITH 2 D + 1 VERTICES 9 For a polytope P that contains the origin in its relative interior, the polar polytope P ∗ is defined as P ∗ = (cid:8) y ∈ R d (cid:12)(cid:12) x · y (cid:54) x in P (cid:9) , where · denotes the dot product of vectors. If the interior of P does not contain theorigin, we translate the polytope so that it does. This translation does not affectthe combinatorics of P or P ∗ .Here we collect some basic information on duals of cyclic polytopes. Lemma 16 (Duals of cyclic polytopes, [15]) . Let P be a cyclic d -polytope on n > d vertices. Then the following statements hold. (i) P ∗ is a simple d -polytope. (ii) For even d , every facet of P ∗ is the dual of C ( n − , d − . In the particular case of the dual C ∗ ( n,
4) of C ( n, n ( n −
3) verticesand n facets, with each facet being C ∗ ( n − , n − n -wedge for every n (cid:62) n -polygon Q n on the hyperplane R × { } of R , constructing a cylinder Q n × R ⊂ R , and cutting the cylinder with two distinct hyperplanes through anedge E × { } of Q n × { } . It follows that the n -wedge is a 3-polytope with 2 n − n + 1 2-faces: two n -gons, two triangles, and n − The result of Xue [19].
Let f k ( F, P ) be the number of k -faces in P con-taining the face F . The result [19, Prop. 3.1], originally designed for d -polytopeswith up to 2 d vertices, is key to our work. Its proof in [19] actually establishessomething slightly more general. Let dim P denote the dimension of a polytope P . Proposition 17 ([19, Prop. 3.1]) . Let d (cid:62) and let P be a d -polytope. In addition,suppose that r (cid:54) d +1 is given and that S := ( v , v , . . . , v r ) is a sequence of verticesin P . Then the following holds. (i) There is a sequence F , F , . . . , F r of faces of P such that each F i has di-mension d − i + 1 and contains v i , but does not contain any v j with j < i . (ii) If deg F i ( v i ) (cid:54) F i − for each i ∈ [1 , min { r, d − } ] , then the numberof k -faces of P that contain at least one of the vertices in S is bounded frombelow by r (cid:88) i =1 φ k − (deg F i ( v i ) , dim F i − for each k (cid:62) . Proof sketch.
The proof of (i) is as in the proof of [19, Prop. 3.1]. The extensionto the case r = d + 1 is easy; take F d +1 = { v d +1 } . For the veracity of (ii) observethat the number f k ( v i , F i ) of k -faces in F i containing v i equates to the number of( k − F i /v i of F i at v i . It follows that f ( F i /v i ) = deg F i ( v i ) (cid:54) F i − , D -POLYTOPES WITH 2 D + 1 VERTICES and so the lower bound φ k − (deg F i ( v i ) , dim F i −
1) of Theorem 2 applies to f k − ( F i /v i ).In case that i = d or d + 1, then dim F i (cid:54)
1, and for each k (cid:62) f k ( v i , F i ) = 0 = φ k − (deg F i ( v i ) , dim F i − . Consequently, for each k (cid:62) r (cid:88) i =1 f k ( v i , F i ) = r (cid:88) i =1 f k − ( F i /v i ) (cid:62) r (cid:88) i =1 φ k − (deg F i ( v i ) , dim F i − . The sketch of the proof is now complete. (cid:3)
Several scenarios of Proposition 17 turn out to be very helpful, and so we gatherthem in the next corollary.
Corollary 18.
Let P be a d -polytope with d + 1 vertices, let r (cid:54) d , and let S be asequence of r vertices in P . Then the number of k -faces of P that contain at leastone of the vertices in S is at least (i) φ k − (deg P ( v ) , d −
1) + r (cid:88) i =2 (cid:18) d − i + 1 k (cid:19) , for any chosen vertex v ∈ S. This expression is bounded from below by (ii) (cid:18) dk (cid:19) + (cid:18) d − k (cid:19) − (cid:18) d − k (cid:19) + r (cid:88) i =2 (cid:18) d − i + 1 k (cid:19) , if v is nonsimple , and in any case by (iii) r (cid:88) i =1 (cid:18) d − i + 1 k (cid:19) . . Polytopes with few edges
The case k = d − d -polytopes with d + 2 vertices was characterised in[10, Chap. 10]. Such polytopes have at least φ ( d + 2 , d ) = (cid:0) d (cid:1) − φ ( d + 2 , d ) edges is the triplex M (2 , d − d -polytopes with d +3 vertices. We know thatthe only one with exactly φ ( d + 3 , d ) edges is the triplex M (3 , d − φ ( d + 3 , d ) + 1 edges are the ( d − − fold pyramid overthe pentagon, and the ( d − − fold pyramid over the tetragonal antiwedge; this iseasily deduced from Gale diagram techniques like those about to be presented (seealso [18, Theorem 3.3]). We shall completely characterise all such polytopes with φ ( d + 3 , d ) + 2 edges, at least in terms of their Gale diagrams.The structure of polytopes with φ ( d + 3 , d ) + 3 or more edges is of course morecomplicated; we refer to [7] for more details. Complete catalogues of d -polytopeswith d + 3 vertices have been constructed only for d (cid:54)
6; see [6].
LOWER BOUND THEOREM FOR d -POLYTOPES WITH 2 D + 1 VERTICES 11 A vertex v is pyramidal in a polytope P if P is a pyramid with apex v ; otherwisethe vertex is nonpyramidal . Alternatively, a vertex v is pyramidal in P if P is theconvex hull of v and a ( d − v , the base of thepyramid.We assume familiarity with Gale diagrams as in [10, § d + s vertices,the transforms of nonpyramidal vertices are located on the unit sphere in R s − ,while pyramidal vertices are transformed to the origin. Different vertices may betransformed to the same point.We state a number of fundamental properties of Gale diagrams. We start withthe fact that any open hemisphere must contain the transforms of at least twovertices. A coface of a polytope is a collection of points in its Gale diagram whoseconvex hull contains the origin in its relative interior. And a collection of verticesin a polytope is the vertex set of a face if, and only if, the transformed points inthe Gale diagram are the complement of a coface. This makes it relatively easyto read off the vertex-facet incidence relations from the Gale diagram; one justhas to identify the minimal cofaces. For example, if two vertices u and v are suchthat their transforms U, V are diametrically opposite, then there is a facet whichcontains all vertices except u and v . On the other hand, if a collection of threeor more vertices have their Gale transforms at the same point on the sphere, thenthey must all be adjacent to one another, since, for every pair of them, the convexhull of the transforms of the remaining vertices will be the same.We are only interested here in polytopes with d + 3 vertices, so the Gale diagramis two-dimensional. This makes it particularly easy to read off the missing edgesfrom the Gale diagram; a missing edge being a pair of nonadjacent vertices. Wewill introduce some temporary vocabulary to streamline our arguments in thissection. Say that two points on a Gale diagram are contiguous if no other pointlies on the short arc between them. This includes the case of two points at thesame location, but not three. Diametrically opposite points are not contiguous;both semicircles between them must contain other points. The point diametricallyopposite a transformed vertex V will be denoted − V ; it need not be the transformof any vertex. The next lemma gathers a number of results that plainly follow fromthe discussion in [10, § Lemma 19.
Let t, u, v . . . be vertices of a d -polytope P with d + 3 vertices, and let T, U, V . . . be their Gale transforms. (i) if V is contiguous with both U and W , then u and w must be adjacent in P . (ii) if v and w are not adjacent, then V and W must be contiguous. If u is another vertex with U contiguous to V (on the other side from W ), D -POLYTOPES WITH 2 D + 1 VERTICES then the Gale transforms of all other vertices must be contained in a closedsemicircle from W to − W . (iii) if v is a simple vertex, then there are precisely two vertices u, w whose trans-forms U, W are contiguous to V , and the transforms of all other verticesare (on the short arc) between − U and − W . (iv) if t, u, v, w are four vertices of P whose transforms T, U, V, W are pairwisecontiguous (in that order), and u, v is a missing edge of P , then the arcfrom W to T (in the given orientation) is the short one. Let P ⊂ R d + d +1 be a d -polytope and P ⊂ R d + d +1 a d -polytope such thattheir affine hulls are skew; two affine spaces are skew if they do not intersect, andno line from one space is parallel to a line from the other. The free join of thepolytopes P and P is the ( d + d + 1)-polytope P := conv( P ∪ P ). The k -facesof the free join are given next; see [10, exercise 4.8.1]. Lemma 20.
Let P and P be two polytopes, and let P be the free join of P and P . The k -faces of P are precisely the free joins of a face F of P and a face F of P such that dim F + dim F + 1 = k . Lemma 21.
Let P be a nonpyramidal d -polytope with d + 3 vertices and exactly φ ( d + 3 , d ) + 2 edges. Then its standard contracted Gale diagram is one of the sixindicated in Fig. 2. The affix d − indicates that d − vertices of P have theirtransforms located at the same point on the circle. (i) The polytope represented by Fig. 2(i) exists only in dimension d = 3 ; it has7 facets, and is the dual of Σ(3) ; (ii) The polytope in Fig. 2(ii) exists in any dimension d (cid:62) ; it has d + 1 facets, and is the dual of a pentasm; (iii) The polytope represented by Fig. 2(iii) exists in any dimension d (cid:62) ; ithas d facets; (iv) The polytope represented by Fig. 2(iv) exists in any dimension d (cid:62) ; it has d − facets; (v) The polytope represented by Fig. 2(v) exists only in dimension d = 4 ; it has facets, and corresponds to the second diagram in [10, Fig. 6.3.4]; (vi) The polytope represented by Fig. 2(vi) exists only in dimension d = 5 ; ithas facets, and is the free join of two quadrilaterals.Proof. The graph of P has exactly four missing edges. We consider its comple-ment graph, which is a graph containing just four edges (and a number of isolatedvertices). It follows from Lemma 19(i)-(ii) that these cannot form a cycle.First, suppose that the complement of the graph contains a path of length three,but not four. Label the vertices in this path, in order, as t, u, v, w . Since t, u and v, w are missing edges, all points in the Gale diagram must be between − U and − V . Since u, v is a missing edge, T and W must lie in a closed semicircle notcontaining U or V . The existence of an additional two vertices vertices constituting LOWER BOUND THEOREM FOR d -POLYTOPES WITH 2 D + 1 VERTICES 13 ( ii ) ( iii )( iv ) d − ( vi )( v )( i ) d − d − Figure 2.
Gale diagrams of nonpyramidal d -polytopes with d + 3vertices and φ ( d + 3 , d ) + 2 edges.a fourth missing edge implies that T, U, V, W all lie in a closed semicircle, whichensures that T and W are diametrically opposite. It follows that there are only sixvertices altogether, so d = 3, and P is in fact the dual of Σ(3). This is case (i). Weremark that the two other transformed points must be strictly between − U and − V , otherwise either t or w would belong to an additional missing edge.Next, suppose they form a path of length four. We label the five vertices as t, u, v, w, x so that each successive pair is a missing edge of P . Applying Lemma 19(ii)to both the missing edges t, u and w, x ensures that the Gale transforms of all othervertices are located at − V . The next question is how many diametrically oppositepairs there are, amongst T, U, V, W, X ? If there are none, we are in case (ii), whichis easily checked to be the dual of a pentasm: the d − d (cid:62) d = 3,these are legitimate Gale diagrams, but (ii) would represent a tetragonal antiwedge,which has five missing edges, while (iii) would represent a simplicial prism, whichhas six missing edges.)Every facet of P has either d or d + 1 vertices, so every minimal proper cofacecontains either two or three transformed vertices. This makes it easy to read offthe minimal cofaces from the Gale diagram. In case (iii), there are 2 d of them, d − d + 1 which contain three points. In case (iv), there are 2 d − d diameters and d − u, v and v, w the corresponding missing edges. Then thetransforms of every other vertex must lie on the arc from − U to − W . Let a, b D -POLYTOPES WITH 2 D + 1 VERTICES be another missing edge and suppose that one (or both) of A, B is not located at − U . Then all transformed points other than A, B must in the closed semicirclefrom U to − U ; in particular all transformed points other than A, B, U, V, W mustbe located at − U . Repeating this argument for the other missing edge, we see thatthere can only be seven points altogether, with two located at − U and two locatedat − W . The only possibility is (v), so d = 4. For the record, its facets are fourquadrilateral pyramids and four simplices.Finally suppose that the complement of the graph contains four disjoint edges.For each such pair of vertices, the transforms of all other vertices must lie in aclosed semicircle. Similar reasoning shows there can only be these eight vertices, so d = 5 and the Gale diagram (vi) is the only possibility. (cid:3) We next establish the main result for the special case k = d −
2: a d -poytope with2 d + 1 vertices and no more ridges than the d -pentasm must itself be a pentasm,unless it has d + 2 facets or is 3-dimensional. Proposition 22.
Let P denote any d -polytope. (i) If P has at least d + 3 vertices, at most φ ( d + 3 , d ) + 2 edges and exactly d + 1 facets, then P must be the dual of a pentasm or the dual of Σ(3) . (ii) A d -polytope with d + 1 vertices, at least d + 3 facets, and no more ( d − -faces than the pentasm must be a pentasm or Σ(3) .Proof. (i) If d = 3, the hypotheses imply that f ( P ) = (6 , , d (cid:62) d -polytope with 2 d +1 or more vertices has at least d (2 d +1) > φ ( d +3 , d )+2edges. And a d -polytope with d + s vertices, where d (cid:62) s (cid:62)
4, has at least φ ( d + s, d )edges. The definition (2) yields that φ ( d + s, d ) is an increasing function of s , andso φ ( d + s, d ) (cid:62) φ ( d + 4 , d ) > φ ( d + 3 , d ) + 2for s (cid:62)
4. Thus a d -polytope with d + s vertices, where d (cid:62) s (cid:62)
4, has more than φ ( d + 3 , d ) + 2 edges. This leaves only the case when P has exactly d + 3 vertices.If P has φ ( d +3 , d ) edges or φ ( d +3 , d )+1 edges, then it is a multifold pyramid overa simplicial 3-prism, a pentagon, or a tetragonal antiwedge, and these respectivelyhave d + 2, d + 3, or d + 3 facets, less than 2 d + 1.Lemma 21 informs us that, apart from pentasms and Σ(3), every nonpyramidal d -polytope with d + 3 vertices and φ ( d + 3 , d ) + 2 edges has strictly less than 2 d + 1facets. Forming a k -fold pyramid increases both the dimension and the numberof vertices by k . The only possibility to have exactly 2 d + 1 facets is then to benonpyramidal, and to be in case (i) or (ii) of the preceding lemma.(ii) Since f d − ( P m ( d )) = ( d + 5 d −
2) = φ ( d + 3 , d ) + 2, the conclusion followsby duality from (i). (cid:3) LOWER BOUND THEOREM FOR d -POLYTOPES WITH 2 D + 1 VERTICES 15 Main result
Recall our aim of showing that for d (cid:62) d -pentasms and d -polytopes with d + 2facets are the extremal d -polytopes with 2 d + 1 vertices. Lemma 23.
Fix t (cid:62) and d (cid:62) . Let P be a d -polytope with at least d +1 verticesand at least d + 3 facets. If P is a t -fold pyramid over a simple polytope, then (i) f k ( P ) (cid:62) f k ( P m ( d )) , if k = 0 or k = d − ; (ii) f k ( P ) > f k ( P m ( d )) , if k ∈ [1 , d − .Proof. The statement (i) is obvious from the hypotheses. Now fix k ∈ [1 , d − t = 0. In this case, P is a simple d -polytope withat least d +3 facets. By the lower bound theorem for simple polytopes (Theorem 1),we obtain that f k ( P ) (cid:62) (cid:18) dk + 1 (cid:19) f d − − (cid:18) d + 1 k + 1 (cid:19) ( d − − k ) . Since f d − ( P ) (cid:62) d + 3 we also have that f k ( P ) (cid:62) (cid:18) dk + 1 (cid:19) ( d + 3) − (cid:18) d + 1 k + 1 (cid:19) ( d − − k )= (cid:18) dk + 1 (cid:19) ( d + 3) − (cid:18) d + 1 k + 1 (cid:19) ( d − k ) + (cid:18) d + 1 k + 1 (cid:19) = (cid:18) dk + 1 (cid:19) ( d + 3) − (cid:18) dk + 1 (cid:19) ( d + 1) + (cid:18) d + 1 k + 1 (cid:19) by (3)= (cid:18) d − k (cid:19) + (cid:18) d − k + 1 (cid:19) + (cid:18) dk + 1 (cid:19) + (cid:18) d + 1 k + 1 (cid:19) by (4)= (cid:18) d − k + 1 (cid:19) + f k ( P m ( d )) by (7) > f k ( P m ( d )) . Now consider the case d = 3. We have established this for t = 0, and thecase t = 2 does not arise, because a tetrahedron has 4 < d + 1 vertices. For t = 1, P is a pyramid over an n -gon, with n (cid:62)
6, in which case, we have that f ( P ) = 2 n >
11 = f ( P m (3)). Hence we have the basis case for an inductiveargument on d . Assume d (cid:62) t (cid:62)
1. Then P is a pyramid over a ( d − F , itselfa ( t − F has at least 2( d −
1) + 2vertices and at least ( d −
1) + 3 facets, and our induction hypothesis applies to F .From P being a pyramid over F it follows that(9) f k = f k ( F ) + f k − ( F ) . In case k = 1, f ( F ) > f ( P m ( d − d − + ( d − − f ( F ) = f ( P ) −
1. Thus, from (9) we obtain that f ( P ) > ( d − + ( d − − d = d + d − f ( P m ( d )) . D -POLYTOPES WITH 2 D + 1 VERTICES In case k ∈ [2 , d − f k ( P ) = f k ( F ) + f k − ( F ) > f k ( P m ( d − f k − ( P m ( d − f k ( P m ( d )) . The last equation is a straightforward application of (4) to (7). Consequently, wehave that f k ( P ) > f k ( P m ( d )) for t (cid:62) k ∈ [1 , d − (cid:3) A corollary of Lemma 23 is the following.
Corollary 24.
Let d (cid:62) , and let P be a d -polytope with at least d + 1 vertices. If P is a multifold pyramid over a simple polytope such that f k ( P ) (cid:54) f k ( P m ( d )) forsome k ∈ [1 , d − , then P has d + 2 facets, in which case P is a t -fold pyramidover ∆( r, s ) for suitable r, s, t (in particular, d = r + s + t ). By virtue of Corollary 24, an extremal d -polytope P with 2 d + 1 vertices otherthan a multifold pyramid over ∆( r, s ) must have a nonpyramidal, nonsimple vertexin P . Our strategy is to divide the problem of finding the d -polytopes with 2 d + 1vertices and minimum number of faces into two parts: first for polytopes that haveat least one nonpyramidal, nonsimple vertex, and then for polytopes in which everyvertex is either pyramidal or simple. In the first case, the only minimisers turn outto be pentasms; in the second case, polytopes with d + 2 facets also come into play.Our main result has an inductive step, which runs more smoothly if we presentthe low-dimensional cases first. While most of the following is known, we state itfor the sake of clarity. Lemma 25.
Consider a d -polytope P with d + 1 vertices, where d (cid:54) . (i) Let d = 3 . If P is Σ(3) or a pentasm, then f ( P ) = (7 , , . Otherwise, f ( P ) > and f ( P ) > . (ii) Let d = 4 . If P is ∆(2 , , then f ( P ) = (9 , , , . If P is a pentasm, then f ( P ) = (9 , , , . Otherwise, f ( P ) (cid:62) , f ( P ) (cid:62) , and f ( P ) (cid:62) . (iii) Let d = 5 . If P is a pentasm, then f ( P ) = (11 , , , , . Otherwise, f ( P ) (cid:62) , f ( P ) (cid:62) , f ( P ) (cid:62) , and f ( P ) (cid:62) .Proof. (i) This is an easy exercise from Steinitz’s theorem. One may also consultcatalogues such as [3, Fig. 4].(ii) The f -vectors of ∆(2 ,
2) and the 4-pentasm are easy to verify. Any 4-polytopewith f = 9 and f = 6 must have that f (cid:62)
18 and f (cid:54)
15, and by Euler’s relationwe know that f = f + 3. The only possibility is the simple polytope ∆(2 , P is any 4-polytope with 9 vertices other than these two, then clearly f (cid:62)
7, andTheorem 11 implies that f (cid:62)
20. Thus f = f + f − (cid:62) f -vector(9 , , , f -vector of the pentasm is clear. Suppose P has 11 vertices butis not a pentasm. Theorem 11 then informs us that f (cid:62)
30, and d being primemeans f (cid:62) LOWER BOUND THEOREM FOR d -POLYTOPES WITH 2 D + 1 VERTICES 17 Next we show that f (cid:62)
25; the proof depends on the value of f . If f = 8,Proposition 22(ii) ensures that f (cid:62)
25. We claim that if f = 9, then in fact f (cid:62)
26. Thanks to [17, Theorem 19], the only 5-polytope with 9 vertices and 25or fewer edges is the triplex M (4 , (cid:54) = 11 facets. The claim followsby duality. Next we claim that if f = 10, then f (cid:62)
27. Again, thanks to [17,Theorem 19], the only 5-polytope with 10 vertices and 26 or fewer edges is thesimplicial prism M (5 , (cid:54) = 11 facets. The claim follows by duality.In the case that f (cid:62)
11, we have 2 f (cid:62) f (cid:62)
55, so f (cid:62)
28. So we have that f (cid:62)
25 in all cases.We next claim that f − f (cid:62)
17. This is clear if f = 8 , ,
10 or 11. If f (cid:62) f − f (cid:62) f − f (cid:62)
18 as required. Finally f = f + f − f − f + 2 (cid:62)
30 + 17 −
11 + 2 = 38 . This completes the proof of the lemma. (cid:3)
We remark that a 5-dimensional capped prism has f = 11 and f = 30, whilea pyramid over Σ(4) has f -vector (11 , , , , f , f and f are tight. However the bound for f is not. It can also beproved that there is no 5-polytope with f = 11 and f = 38, but the proof of thisis long, and its inclusion would be an unnecessary digression from the aim of thispaper.The next step in our journey is to show that our minimising polytopes have veryrestricted facial structure. Lemma 26.
Fix d (cid:62) , and let P be a d -polytope with d + 1 vertices and witha nonpyramidal, nonsimple vertex. Suppose f k ( P ) (cid:54) f k ( P m ( d )) for some k ∈ [2 , d − . If F is a facet avoiding at least one nonpyramidal nonsimple vertex, theneither (i) F is a triplex M (2 , d − and every vertex outside F has degree at most d + 1 ; or (ii) F is a simplicial d -prism M ( d − , and every vertex outside F has degreeat most d + 1 ; or (iii) F contains exactly d − vertices, one (or more) of which is nonpyramidaland nonsimple in F ; consequently F has at least d +2 facets (i.e. F contains d + 2 ridges of P ); or (iv) F is a simplex, but there is another facet F (cid:48) that is not a simplex and alsoavoids at least one nonpyramidal nonsimple vertex; hence F (cid:48) falls in one ofthe previous cases (i)–(iii).Proof. Let F be a facet of P not containing a nonpyramidal nonsimple vertex.The facet F is not the base of a pyramid, so we may suppose that F has exactly d − d + 2 − r vertices with r ∈ [2 , d + 1]. There are r vertices outside F ,say S := { v , v , . . . , v r } , none of them pyramidal. Without loss of generality, D -POLYTOPES WITH 2 D + 1 VERTICES suppose that v has the largest possible degree among the vertices in S ; clearly v is nonsimple.(i)-(ii) Consider the case r ∈ [3 , d ]. Since d + 2 − r (cid:54) d −
1, it follows fromTheorem 2 that f k ( F ) (cid:62) φ k ( d − d + 2 − r, d −
1) for k ∈ [1 , d − k -faces in P consist of the k -faces of F plus the k -faces containing at least one of the verticesin S . Corollary 18(ii) then informs us that for each k ∈ [2 , d − f k ( P ) (cid:62) f k ( F ) + φ k − (deg P ( v ) , d −
1) + r (cid:88) i =2 (cid:18) d − i + 1 k (cid:19) (cid:62) φ k ( d − d + 2 − r, d −
1) + φ k − ( d + 1 , d −
1) + r (cid:88) i =2 (cid:18) d − i + 1 k (cid:19) (10) = (cid:18) dk + 1 (cid:19) + (cid:18) d − k + 1 (cid:19) − (cid:18) r − k + 1 (cid:19) + (cid:18) dk (cid:19) + (cid:18) d − k (cid:19) − (cid:18) d − k (cid:19) + r (cid:88) i =2 (cid:18) d − i + 1 k (cid:19) = (cid:18) d + 1 k + 1 (cid:19) + (cid:18) dk + 1 (cid:19) − (cid:18) r − k + 1 (cid:19) − (cid:18) d − k (cid:19) + r (cid:88) i =2 (cid:18) d − i + 1 k (cid:19) = f k ( P m ( d )) − (cid:18) r − k + 1 (cid:19) + r (cid:88) i =4 (cid:18) d − i + 1 k (cid:19) = f k ( P m ( d )) − (cid:18) r − k + 1 (cid:19) + r − (cid:88) j =0 (cid:18) d − − jk (cid:19) (cid:62) f k ( P m ( d )) − (cid:18) r − k + 1 (cid:19) + r − (cid:88) j =0 (cid:18) r − − jk (cid:19) (11) = f k ( P m ( d )) − (cid:18) r − k + 1 (cid:19) + r − (cid:88) j =0 (cid:18) jk (cid:19) = f k ( P m ( d )) (cid:62) f k ( P )where at the end we used (5). Clearly none of the inqualities can be strict. Thisyields immediately that F must be a triplex, and that the vertex v has degreeprecisely d + 1. Equality in (11) forces the conclusion that r = 3 or r = d . Equalityin (11) forces the conclusion that r = 3 or r = d . Equality in (10) forces theconclusions that F is a triplex (more precisely, F is M ( d + 2 − r, r − v has degree precisely d + 1. Since each vertex v i ∈ S is nonpyramidalin P , and v has the largest degree amongst them, we have deg P ( v i ) (cid:54) d + 1, foreach i . This settles the case r ∈ [3 , d ], putting F into either case (i) or case (ii).(iii) Next we look at the case r = 2, meaning that F has 2 d − S = { v , v } . We want to show that F falls into case (iii); let us consider thepossibility that it does not: then every vertex of F is pyramidal or simple therein. LOWER BOUND THEOREM FOR d -POLYTOPES WITH 2 D + 1 VERTICES 19 If this is the case, then Lemma 12 ensures that F is a t -fold pyramid over ∆( m, n ),where m + n + t = d −
1. Without loss of generality, assume that m (cid:54) n . Clearly f ( F ) = ( m + 1)( n + 1) + t = 2 d −
1, which implies that mn = d −
1. This precludesthe possibility that m = 1. Hence 2 (cid:54) m (cid:54) n .Every ridge R of P that is contained in F is either ∆ t − ( m, n ), ∆ t ( m − , n ), or∆ t ( m, n −
1) (Remark 13), and so has at least mn + m + t vertices. For i = 1 , F i be a facet of P containing v i but not v − i . We claim that each F i is apyramid with apex v i ; we just write the details for i = 1. Let R be an arbitraryridge contained in F but not containing v . Clearly R ⊂ F and R ⊂ F , whichforces R = F ∩ F . Thus R is the unique ridge of F not containing v , making F a pyramid over R , and v is then adjacent to every vertex in R . Likewise, F is a pyramid with apex v . So v and v have at least 2( mn + m + t ) edgesrunning into F . This is ( m − n + 1) + t more than the number of vertices in F , so ( m − n + 1) + t vertices in F must be adjacent to both v and v . Just t vertices in F are nonsimple in F , so at least ( m − n + 1) simple vertices in F arenonsimple in P ; let us fix one such vertex u , and note that u is both nonsimple andnonpyramidal in P . Choose a ridge R (cid:48) in F of the form ∆ t ( m − , n ) that avoids u . Then R (cid:48) avoids n + 1 vertices of F altogether, and so does F (cid:48) , the other facetcorresponding to R (cid:48) . In particular, f ( F (cid:48) ) is either 2 d − n or 2 d − n −
1, dependingon whether it contains one or both v i . Write f ( F (cid:48) ) as d − d + 2 − r (cid:48) . Then2 d − n (cid:54) d −
2, implying that r (cid:48) ∈ [3 , d + 1].Since F (cid:48) avoids u , a nonpyramidal nonsimple vertex, the preceding parts showthat r (cid:48) cannot be in [4 , d − f ( F (cid:48) ) is either d (where r (cid:48) = d + 1), d + 1(where r (cid:48) = d ), or 2 d − r (cid:48) = 3). Recalling that mn = d −
1, the onlyinteger solution for f ( F (cid:48) ), with the constraint 2 (cid:54) m (cid:54) n , is m = n = 2 and d = 5,contradicting our assumption that d (cid:62)
6. Thus F falls into Part (iii) after all. Asmentioned in the introduction, this means F has at least ( d −
1) + 3 facets.(iv) Finally, assume that r = d + 1; that is, F is a simplex. We can also assumethat every facet not containing v is a simplex.We claim that there is a ridge R in P that is the intersection of two facets F (cid:48) and F (cid:48)(cid:48) , neither of which contains v . Suppose otherwise. We work on the dualpolytope P ∗ of P ; let σ be an anti-isomorphism from the face lattice of P to theface lattice of P ∗ . Then every ridge of P would belong to a facet containing v .Dually, this means that every edge in P ∗ has an endvertex in the facet σ ( v ) of P ∗ associated with v , and so no two vertices could exist outside σ ( v ). As a result, P ∗ would be a pyramid with base σ ( v ), whose dual statement is that v would bepyramidal at P , a contradiction. Thus the claims holds.The facets F (cid:48) and F (cid:48)(cid:48) are both simplices, and F (cid:48) ∪ F (cid:48)(cid:48) contains d + 1 vertices.Denote by S (cid:48) := { v (cid:48) , v (cid:48) , . . . , v (cid:48) d } the vertices outside F (cid:48) ∪ F (cid:48)(cid:48) ; we may choose v (cid:48) = v .The k -faces in P include the k -faces of F (cid:48) ∪ F (cid:48)(cid:48) and the k -faces containing at least one D -POLYTOPES WITH 2 D + 1 VERTICES of the vertices in S (cid:48) . Corollary 18(ii) gives the following inequalities for k ∈ [2 , d − f k ( P ) (cid:62) f k ( F (cid:48) ∪ F (cid:48)(cid:48) ) + (cid:18) dk (cid:19) + (cid:18) d − k (cid:19) − (cid:18) d − k (cid:19) + d (cid:88) i =2 (cid:18) d − i + 1 k (cid:19) (cid:62) (cid:18) dk + 1 (cid:19) + (cid:18) d − k (cid:19) + (cid:18) dk (cid:19) + (cid:18) d − k (cid:19) − (cid:18) d − k (cid:19) + d (cid:88) i =2 (cid:18) d − i + 1 k (cid:19) = (cid:18) dk + 1 (cid:19) + (cid:18) d − k (cid:19) + (cid:18) d − k − (cid:19) + d (cid:88) i =1 (cid:18) d − i + 1 k (cid:19) (by (4))= (cid:18) dk + 1 (cid:19) + (cid:18) d − k (cid:19) + (cid:18) d − k − (cid:19) + (cid:18) d + 1 k + 1 (cid:19) (by (5))= f k ( P m ( d )) + (cid:18) d − k − (cid:19) (by (7)) > f k ( P m ( d )) (since k ∈ [2 , d − . This shows that not every facet avoiding v can be a simplex. Hence the case (iv)follows, and with it, the proof of the lemma. (cid:3) Lemma 27 is buried in the construction in [10, 10.4.1]; we elucidate the detailshere. The non-existence of simple 4-polytopes with six, seven or ten vertices is wellknown, [10, 10.4.2] or [16, Lemma 2.19].
Lemma 27.
For a fixed f , the minimum value of f ( P ) , amongst all 4-polytopeswith f vertices, is the unique integer n satisfying (cid:0) n − (cid:1) (cid:54) f (cid:54) (cid:0) n − (cid:1) − . If f (cid:62) , then there is a simple 4-polytope with f vertices and this minimum numberof facets.Proof. For n (cid:62)
2, the intervals (cid:2)(cid:0) n − (cid:1) , (cid:0) n (cid:1) − (cid:3) are obviously disjoint, and theirunion is all the natural numbers. So given f , there is a unique integer n satisfyingthe given inequality. First we claim that no 4-polytope with f vertices has fewerthan n facets. Refer to [10, § f (cid:54) f ( f − , which (given that f is positive) is equivalent to f (cid:62) ( √ f + 9 + 3). If f > (cid:0) n − (cid:1) −
1, then f > (cid:16)(cid:112) (2 n − + 3 (cid:17) = n − , which settles the claim.Next we prove by induction on n (cid:62) (cid:0) n − (cid:1) (cid:54) f (cid:54) (cid:0) n − (cid:1) − f (cid:62) f vertices and n facets.In the base case n = 7 we have 10 (cid:54) f (cid:54)
14. This case was settled by Br¨uckner[5], who completely classified the simple 4-polytopes with 7 facets. They are thesimplicial 4-prism with one vertex truncated [5, Fig. 9], the polytope ∆(2 ,
2) withone vertex truncated [5, Fig. 10 or 10a], the polytope ∆(1 , ,
2) (i.e. the Minkowskisum of a square and a triangle) [5, Fig. 11], the polytope ∆(2 ,
2) with one edge
LOWER BOUND THEOREM FOR d -POLYTOPES WITH 2 D + 1 VERTICES 21 truncated [5, Fig. 12], and C ∗ (7 ,
4) [5, Fig. 13]. They have respectively 11, 12, 12,13, and 14 vertices.Suppose that n (cid:62)
8. Then (cid:18) n − (cid:19) < (cid:18) n − (cid:19) − < (cid:18) n − (cid:19) − < (cid:18) n − (cid:19) − . Therefore, by the induction hypothesis, there are two simple 4-polytopes with n − f = (cid:0) n − (cid:1) − f = (cid:0) n − (cid:1) − f = (cid:0) n − (cid:1) and (cid:0) n − (cid:1) + 1, respectively.Because C ∗ ( n − ,
4) contains ( n − n −
4) = (cid:0) n − (cid:1) − n − F with n − m vertices of F , for 1 (cid:54) m (cid:54) n −
4, asin Lemma 5(v), will give us a simple polytope with n facets and (cid:0) n − (cid:1) − m + 2vertices. This establishes the conclusion for (cid:18) n − (cid:19) + 2 (cid:54) f (cid:54) (cid:18) n − (cid:19) − n − (cid:18) n − (cid:19) − , thereby completing the proof of the theorem. (cid:3) We require a definition and a result of [10, § v is beneath a facet ofa polytope if v belongs to the open halfspace that is determined by the supportinghyperplane for that facet and contains the interior of the polytope. Theorem 28 ([10, Thm. 5.2.1]) . Let P and P (cid:48) be two d -polytopes in R d , and let v be a vertex of P (cid:48) such that v (cid:54)∈ P and P (cid:48) = conv( P ∪ { v } ) . Then (i) a face F of P is a face of P (cid:48) if and only if there exists a facet J of P suchthat F ⊆ J and v is beneath J ; (ii) if F is a face of P then F (cid:48) := conv( F ∪ { v } ) is a face of P (cid:48) if(a) either v is in the affine full of F ;(b) or among the facets of P containing F there is at least one such that v is beneath it and at least one such that v is beyond it.Moreover, each face of P (cid:48) is of exactly one of the above three types. We show that the d -pentasm is the unique minimiser of the number of k -facesamong the d -polytopes with 2 d + 1 vertices and at least one vertex that is bothnonpyramidal and nonsimple. Theorem 29.
Let d (cid:62) . Let P be a d -polytope with d + 1 vertices, at least one ofwhich is both nonpyramidal and nonsimple. Suppose f k ( P ) (cid:54) f k ( P m ( d )) for some k ∈ [1 , d − . Then P is a pentasm.Proof. By induction on d . The base case d = 5 is contained in Lemma 25, so weassume d (cid:62) k = d − k = 1for d (cid:62) d -polytope with 2 d + 1 verticesand at most d + d − d -pentasm, i.e. to Theorem 11. So we assume2 (cid:54) k (cid:54) d − D -POLYTOPES WITH 2 D + 1 VERTICES Amongst the facets in P that avoid at least one nonpyramidal, nonsimple vertex,choose F with a maximum number of vertices. By Lemma 26, F has exactly d − d + 2 − r vertices with r = 2 , d . There are r vertices outside F .We begin with the case r = 2, namely Lemma 26(iii). Then F has a nonpyrami-dal, nonsimple vertex and 2( d −
1) + 1 vertices. It follows that the two vertices v and v outside F are nonpyramidal. They must be adjacent, and one of them mustbe adjacent to at least d vertices in F . Without loss of generality, suppose that v isnonsimple and its degree in P is not smaller than that of v . From Proposition 17,we get a short sequence F , F of faces such that F = P , F is a facet, v i ∈ F i ,and v (cid:54)∈ F . Any k -face of P falls into one of four disjoint groups: either it • is contained in F , or • contains v , or • contains v and is contained in F , or • contains v but not v , and is not contained in F .Denote by n the number of k -faces in this last group. Proposition 17, together withour hypothesis, now yields the following estimates: f k ( P ) = f k ( F ) + φ k − (deg P ( v ) , d −
1) + (cid:18) d − k (cid:19) + n (12) (cid:62) f k ( P m ( d − (cid:18) dk (cid:19) + (cid:18) d − k (cid:19) − (cid:18) d − k (cid:19) + (cid:18) d − k (cid:19) + 0= (cid:18) dk + 1 (cid:19) + (cid:18) d − k + 1 (cid:19) + (cid:18) d − k (cid:19) + (cid:18) dk (cid:19) + (cid:18) d − k (cid:19) − (cid:18) d − k (cid:19) + (cid:18) d − k (cid:19) = (cid:18) d + 1 k + 1 (cid:19) + (cid:18) dk + 1 (cid:19) + (cid:18) d − k (cid:19) = f k ( P m ( d )) (cid:62) f k ( P ) . Clearly none of these inequalities can be strict, which leads to the following con-clusions.(a) f k ( F ) = f k ( P m ( d − v has (again) degree precisely d + 1, and therefore deg P ( v ) (cid:54) d + 1.(c) The vertex v is simple in F .(d) n = 0.From Observation (a) the induction hypothesis kicks in, to inform us that F isa pentasm. Observation (d) means that every k -face containing v but not v mustbe contained in F . In other words, v is the only neighbor of v not contained in F . Combined with Observation (c), we obtain that v is simple in P .Since v has degree d + 1 and v is simple in P , the number of edges of P is f ( F ) + 2 d , the number of edges of P m ( d ). Again Theorem 11 yields that P is apentasm. LOWER BOUND THEOREM FOR d -POLYTOPES WITH 2 D + 1 VERTICES 23 We now consider the cases r = 3 , d . These are respectively cases (ii) and (i) ofLemma 26, so we may assume that every facet avoiding a nonpyramidal nonsimplevertex has at most 2 d − P is either pyramidalor nonpyramidal of degree at most d + 1.If the excess degree ξ of P were d − P would be a d -pentasm by Theorem 11.Therefore, we may assume that ξ ( P ) (cid:62) d −
1. In addition, since d (cid:62)
6, the parityof ξ (Remark 6) ensures that ξ ( P ) (cid:62) r = 3, then some facet F avoiding a nonpyramidal, nonsimple vertex hasexactly 2 d − d -prism (Lemma 26(ii)),and so no vertex in P is pyramidal, whence every vertex of P is either simple or hasexcess degree one. Thus P contains at least six nonsimple vertices (as ξ ( P ) (cid:62) F as u , u , . . . , u d − , x , x , . . . , x d − so that the sets u , u , . . . , u d − and x , x , . . . , x d − respectively form two ( d − u i x i is an edge E i for i = 1 , . . . , d − F , without loss of general-ity say u , must be nonsimple. Denote by R the ( d − F containing u , . . . , u d − , x , . . . , x d − , and by F (cid:48) the other facet of P containing R . Since F (cid:48) has at least 2 d − d + 1 for d (cid:62)
6, but does notcontain the nonsimple vertex u , F (cid:48) must also be a simplicial ( d − u d , x d be the two vertices in F (cid:48) \ F . Because F is a simplicial prism, the linescontaining E , E , . . . , E d − must all be either parallel or concurrent at a pointoutside F by Remark 9. Similarly, as F (cid:48) is a simplicial prism, the lines containing E , . . . E d − , E d must all be either parallel or concurrent at a point outside F (cid:48) . Con-sequently, the same conclusion holds for E , E , . . . , E d . Denote by w the uniquevertex not in F ∪ F (cid:48) .Since P has at least six nonsimple vertices, there must be a nonsimple vertexdifferent from u , u d , x , x d , w . Without loss generality, this vertex is either u or x . Let R (cid:48) denote the ridge containing the edges E , E , . . . , E d − . Then R (cid:48) ⊂ F ;let F (cid:48)(cid:48) be the other facet containing R (cid:48) . Then F (cid:48)(cid:48) avoids a nonsimple vertex ( u or x ) and so, as before, must be a simplicial prism. In particular F (cid:48)(cid:48) containsprecisely two of u d , x d , w . Suppose F (cid:48)(cid:48) contains u d and w . For F (cid:48)(cid:48) to be a simplicialprism, E = u d w must be an edge of it, and the lines containing E , E , . . . E d − , E must all be either parallel or concurrent at a point outside F (cid:48)(cid:48) . But then all the linescontaining E , E , E , . . . , E d , E must be parallel or concurrent at a point outside F (cid:48)(cid:48) . Since E and E d are concurrent at u d , this is impossible. Similarly, the twovertices in F (cid:48)(cid:48) \ R (cid:48) cannot be x d and w ; they must be u d and x d . Since F (cid:48)(cid:48) is asimplicial prism, this implies that u u d and x x d are edges of F (cid:48)(cid:48) , and of P .Thus the convex hull of F ∪ F (cid:48) is a simplicial d -prism, which we will denote by Q . Then P = conv( Q ∪ { w } ). We have that F and F (cid:48) are facets of Q and P . Inthis setting, Theorem 28(i) ensures that w is beneath F and F (cid:48) . Besides, because w has degree at most d + 1 in P , it cannot be beyond any of the simplicial prismfacets of Q . Thus w is beyond only one of the simplex facets of Q , which implies D -POLYTOPES WITH 2 D + 1 VERTICES that P is a capped prism (Section 2.6). But any capped prism has more k -facesthan the pentasm, for k (cid:54) d −
2, thanks to Lemma 15.Now suppose that r = d , i.e. the facet F is a ( d − F is a pyramid with apex u i and base R i for each i ∈ [1 , d −
3] where R i is a ( d − v be anonsimple vertex outside F , and let S denote the set of vertices outside F .Suppose that some u i of F is nonpyramidal in P , say u . Then the other facet F (cid:48) containing R must be a ( d − u (cid:54)∈ F (cid:48) , u is nonpyramidal and nonsimple in P , and F has the largest number ofvertices among all facets avoiding a nonpyramidal, nonsimple vertex in P . Withoutloss of generality, assume v ∈ F (cid:48) ; this is fine because the vertex in S ∩ V ( F (cid:48) ) isnonpyramidal and nonsimple in P . Corollary 18(ii) applied to S \ { v } implies thefollowing inequalities for k ∈ [2 , d − f k ( P ) (cid:62) f k ( F ∪ F (cid:48) ) + d − (cid:88) i =1 (cid:18) d − i + 1 k (cid:19) (cid:62) (cid:18) dk + 1 (cid:19) + (cid:18) d − k (cid:19) + (cid:18) d − k (cid:19) + (cid:18) d − k − (cid:19) + d − (cid:88) i =1 (cid:18) d − i + 1 k (cid:19) = (cid:18) dk + 1 (cid:19) + (cid:18) d − k (cid:19) + (cid:18) d − k (cid:19) + (cid:18) d − k − (cid:19) + (cid:18) d + 1 k + 1 (cid:19) (by (5))= f k ( P m ( d )) + (cid:18) d − k (cid:19) + (cid:18) d − k − (cid:19) > f k ( P m ( d )) . Now assume that every u i is pyramidal at P , and so P is a ( d − d + 4 vertices. We find it more convenient to consider P asa ( d − Q with d + 5 vertices, equivalently as thefree join of a ( d − Q .Lemma 27 gives us a simple 4-polytope Q (cid:48) with f ( Q (cid:48) ) = f ( Q ) = d + 5 (cid:62) f ( Q (cid:48) ) minimal amongst all 4-polytopes with d + 5 vertices. Being simple,we have f ( Q (cid:48) ) = 2 f ( Q (cid:48) ) = 2 f ( Q ) (cid:54) f ( Q ). Euler’s relation then implies that f ( Q (cid:48) ) (cid:54) f ( Q ) as well. Denote by P (cid:48) the free join of a ( d − T and Q (cid:48) .Then, by Lemma 20 we have f k ( P ) = f k ( T ) + f k − ( T ) f ( Q ) + f k − ( T ) f ( Q ) + f k − ( T ) f ( Q ) + f k − ( T ) f ( Q )+ f k − ( T ) (cid:62) f k ( T ) + f k − ( T ) f ( Q (cid:48) ) + f k − ( T ) f ( Q (cid:48) ) + f k − ( T ) f ( Q (cid:48) ) + f k − ( T ) f ( Q (cid:48) )+ f k − ( T )= f k ( P (cid:48) ) > f k ( P m ( d )) , where the last inequality f k ( P (cid:48) ) > f k ( P m ( d )) comes from Lemma 23. This con-cludes this case, and with it, the proof of the theorem. (cid:3) LOWER BOUND THEOREM FOR d -POLYTOPES WITH 2 D + 1 VERTICES 25 An immediate consequence of Lemma 23 and Theorem 29 is the following.
Theorem 30.
Let d (cid:62) . Let P be a d -polytope with d + 1 vertices such that f k ( P ) (cid:54) f k ( P m ( d )) for some k ∈ [1 , d − . (i) If at least one vertex in P is both nonpyramidal and nonsimple, then P isa pentasm. (ii) If every vertex in P is pyramidal or simple, then P is a t -fold pyramid over ∆( r, s ) for some r, s > and t (cid:62) such that d = r + s + t . Combining Corollary 14, Lemma 25, and Theorem 30, we get the conclusion ofTheorem 3.When d is prime, the pentasm is the unique minimiser of f k for all k (cid:54) d − d is composite, Theorem 3 gives us a finite list of candidates for theminimiser of each f k . We conjecture the following more precise conclusion, whichwe have verified for all d (cid:54) d is composite, and k (cid:54) d/
2, then the d -pentasm is the unique minimiserfor f k .(ii) If d = rs is composite, r is the smallest prime factor of d , and k > d/
2, thena ( d − r − s )-fold pyramid over ∆( r, s ) is the unique minimiser of f k ( P ). References
1. D. W. Barnette, The minimum number of vertices of a simple polytope,
IsraelJ. Math. (1971), 121–125. MR 0298553 (45 Pacific J.Math. (1973), 349–354. MR 0328773 (48 Acta Crystallographica Section A (1973), no. 4, 362–371.4. A. Brøndsted, An introduction to convex polytopes , Graduate Texts in Mathe-matics, vol. 90, Springer-Verlag, New York, 1983. MR 683612 (84d:52009)5. M. Br¨uckner, ¨Uber die Ableitung der allgemeinen Polytope und die nach Iso-morphismus verschiedenen Typen der allgemeine Achtzelle (Oktatope),
Ver-handelingen der Koninklijke Akademie van Wetenschappen (1909).6. K. Fukuda, H. Miyata and S. Moriyama. Classification of Oriented Matroids ∼ hmiyata/oriented matroids/7. E. Fusy, Counting d-polytopes with d + 3 vertices, Electron. J. Combin. (2006), no. 1, Research Paper 23, 25 pp.8. D. Gale, Neighborly and cyclic polytopes, Proc. Sympos. Pure Math , vol. 7,1963, pp. 225–232. MR0152944 (27
Concrete Mathematics: Afoundation for computer science , 2nd ed., Addison-Wesley, New York, 1994.MR1397498 (97d:68003) D -POLYTOPES WITH 2 D + 1 VERTICES
10. B. Gr¨unbaum,
Convex polytopes , 2nd ed., Graduate Texts in Mathematics,vol. 221, Springer-Verlag, New York, 2003, Prepared and with a preface by V.Kaibel, V. Klee and G. M. Ziegler. MR 1976856 (2004b:52001)11. S. Klee, E. Nevo, I. Novik, and H. Zheng, A lower bound theorem for cen-trally symmetric simplicial polytopes,
Discrete & Computational Geometry (2019), no. 3, 541–561. MR391854712. S. Klee and I. Novik, Lower bound theorems and a generalized lower boundconjecture for balanced simplicial complexes., Mathematika (2016), no. 2,441–477. MR352133513. P. McMullen, The minimum number of facets of a convex polytope, J. LondonMath. Soc. (2) (1971), 350–354. MR 028599014. P. McMullen and G. C. Shephard, Convex polytopes and the upper bound con-jecture . London Mathematical Society Lecture Note Series, . Cambridge Uni-versity Press, London-New York, 197115. M. A. Perles and G. C. Shephard, Facets and nonfacets of convex polytopes, Acta Math. (1967), 113–145. MR 022397516. G. Pineda-Villavicencio, J. Ugon, and D. Yost, The excess degree of a poly-tope,
SIAM Journal on Discrete Mathematics (2018), no. 3, 2011–2046.MR384088317. , Lower bound theorems for general polytope, European J. Combin. (2019), 27–45. MR389908318. , Polytopes close to being simple, Discrete & Computational Geometry (2020), 200–215, MR411053319. L. Xue, A proof of Gr¨unbaum’s lower bound conjecture for polytopes , arXiv2004.08429, 2020.20. G. M. Ziegler,