A SEPARATION THEOREM FOR TOTALLY-SEWN 4-POLYTOPES

A SEPARATION THEOREM FOR TOTALLY-SEWN 4-POLYTOPES

T. BISZTRICZKY¹ AND F. FODOR²

Abstract. The Separation Problem, originally posed by K. Bezdek in [1], asks for the minimum numbers(O, K) of hyperplanes needed to strictly separate an interior pointOin a convex bodyKfrom all faces ofK. It is conjectured thats(O, K)≤2^dind-dimensional Euclidean space. We prove this conjecture for the class of all totally-sewn neighbourly 4-dimensional polytopes.

1. Introduction

The Gohberg-Markus-Hadwiger Covering Problem is a well-known unsolved prob- lem in convex geometry. It seeks the minimum numberh(K) of smaller homothetic copies of a convex body K (compact convex set in R^d with non-empty interior) whose union covers K. It is conjectured thath(K)≤2^d, and that equality holds only for affined-cubes. The Gohberg-Markus-Hadwiger Covering Problem is solved completely only in two dimensions, cf. [7]. In higher dimensional spaces there are only partial results. For a detailed overview of this topic we refer to [7] and [10].

In this article we consider the Separation Problem which was raised by K. Bezdek [1]. LetKbe a convex body inR^d andO∈intK an interior point. The separation numbers(O, K) ofO inK is defined as the minimum number of hyperplanes that strictly separateO from all faces of K. Bezdek proved in [1] that s(O, K) is equal to the covering number h(K^∗) of the polarK^∗ of K, therefore, it is conjectured thats(O, K)≤2^d.

The evaluation of the separation number seems especially important for poly- topes. There are only a few special classes of polytopes for which this has been accomplished. In particular, we mention here that in [2] and [3] it was shown that s(O, P)<2^d in the case thatP is a cyclicd-polytope.

The celebrated Upper Bound Theorem of McMullen [11] states that among all d-polytopes with a fixed number of vertices, the neighbourlyd-polytopes have the maximum number of facets. Thus, it is natural to investigate s(O, P) for neigh- bourlyd-polytopesP. Since interesting neighbourly polytopes exist only inR^d for

2010Mathematics Subject Classification. 52B11.

Key words and phrases. Gohberg-Markus-Hadwiger covering number, neighbourly polytopes, separation problem, totally-sewn polytopes.

This file is not the final published version of the paper. Published ver- sion: Studia Scientiarum Mathematicarum Hungarica 52 (3), 386–422 (2015) http://dx.doi.org/10.1556/012.2015.52.3.1316.

1Supported in part by an NSERC Discovery Grant.

2Supported by the Hungarian-Mexican Intergovernmental S&T Cooperation Programme under grants T ´ET 10-1-2011-0471 and CONACYT 121158 “Discrete and Convex Geometry”, and by Hungarian OTKA grant No. 75016. This paper was supported by the J´anos Bolyai Research Scholarship of the Hungarian Academy of Sciences.

1

d≥4, it is also natural to first restrict our attention to neighbourly 4-polytopes.

Although a lot of information is known about neighbourly polytopes in general, only a few constructions yield infinite families of such polytopes. The most well-known such construction is the “sewing” operation introduced by Shemer [12]. Starting from a cyclicd-polytope, the sewing procedure of Shemer produces an infinite fam- ily of neighbourlyd-polytopes each of which is obtained from the previous one by adding one new vertex in a suitable way. Neighbourlyd-polytopes obtained from a cyclic d-polytope by a sequence of sewings are called called totally-sewn. Totally- sewn neighbourly 4-polytopes constitute a positive percentage of all neighbourly 4-polytopes with n vertices although this percentage decreases as n increases, cf.

[12]. Moreover, each neighbourly 4-polytope has totally-sewn subpolytopes and this may yield a method of extending the present result to all neighbourly 4-polytopes.

The conjecture thats(O, P)≤9 for neighbourly 4-polytopes was formulated in [4]. This (stronger) conjecture was verified in [6] for those P that have at most ten vertices or that have the property that their vertices form a special configu- ration that resembles a “pentagram”. It was demonstrated in [8] that semi-cyclic 4-polytopes possess this special pentagram property (for the definition of semi- cyclic 4-polytopes see, for example, page 125 in [5]). It was also shown in [6] that there exist totally-sewn neighbourly 4-polytopes that do not have the pentagram property.

In [5], it was proved thats(O, P)≤16 for a special class of totally-sewn neigh- bourly 4-polytopes that have the so-called decreasing universal edge property, cf.

Section 3 of [5]. In this paper we build on the ideas developed in [5] and extend them to the whole class of totally-sewn neighbourly 4-polytopes. Our main result is the Separation Theorem (Theorem 5.1), which asserts that ifP is a totally-sewn neighbourly 4-polytope, then s(O, P) ≤ 16 for any point O in the interior of P. However, it still remains open whether the stronger conjecture [4] thats(O, P)≤9 holds for all totally-sewn neighbourly 4-polytopes.

The rest of the paper is organized as follows. In Section 2, we introduce vertex sewing and vertex types. In Section 3, we examine the universal edge types and the vertex types ofP. We consider the separation of an interior pointO ofP from facets ofP based upon the location (specific in Section 4, and generic in Section 5) ofO inP.

2. Definitions

In this paper, we will work in R⁴. The convex hull and the affine hull of a set X ⊂R⁴ will be denoted by [X] and hXi, respectively. We will use the following notations for the vertices, edges, and facets of a polytope P: V(P), E(P), and F(P), respectively. For x∈ V(P), the vertex figure of P at xwill be denoted by P/x. IfE = [x, y] ∈ E(P), then the quotient polytope P/E is a vertex figure of P/xat the vertex that corresponds toE inP/x.

A 4-dimensional polytope P is neighbourly if for any x, y ∈ V(P), x 6= y, the segment [x, y] is an edge ofP. It is known that 4-dimensional neighbourly poytopes are simplicial. From now on, the symbolP will always denote a convex neighbourly 4-polytope. For basic geometric and combinatorial properties of neighbourly poly- topes we refer to [9] and [12].

An edgeE= [x, y]∈ E(P) isuniversalif for anyz∈ V(P)\ {x, y}, the triangle [E, z] is a 2-face of P. The set of universal edges ofP will be denoted byU(P).

Lemma 2.1 (cf. [12] and [13]). If |V(P)| ≥7, then the following are equivalent:

• E= [x, y]∈ U(P).

• [E, z] is a2-face of P for anyz∈ V(P)\ {x, y}.

• xandy lie on the same side of every hyperplane determined by vertices of P.

• The quotient polytopeP/E is a convex polygon with |V(P)| −2 vertices.

SinceP/E is convex polygon, there is a readily understandable description of all the facets ofP that contain a universal edgeE and how these facets are related to one another. We use this property and the fact that the vertices of any universal edge are not separable to determine the location of the vertices ofP relative to the hyperplanes that containE.

Next, we describe the sewing procedure of Shemer [12] in R⁴. Assume thatP has n vertices and E = [x, y] is a universal edge. LetF(E, P) denote the set of facets that containE. By Lemma 2.1, F(E, P) has n−2 elements. We label the verticesV(P)\ {x, y}={z1, z2, . . . , z_n−2}in such a way that

F(E, P) ={[E, zi, zi+1]|i= 1, . . . , n−2 andz_n−1=z1}.

To keep the notation simple, we denote the edge determined by [E, z_i, z_i+1] inP/E by [z_i, z_i+1].

LetF = [E, zi, zi+1]∈ F(E, P), and letF(E, F, P) =F(E, P)\ {F}be the set of facets ofP that containE and are different fromF. Then there exists a point x∈R⁴ (cf. [12]) which is beyond each facet in F(E, F, P) and beneath all other facets of P. The polytope P = [P, x] is neighbourly (cf. [12]), and it is clear from the location ofxthat V(P) =V(P)∪ {x}. We say that P is obtained fromP by sewing xthrough F(E, F, P). The universal edges ofP were characterized in [12]

as follows:

U(P) =U⁰(P)∪ {[x, x],[x, y]},

whereU⁰(P) ={E⁰∈ U(P)|E⁰∩F =∅or|E⁰∩ {z_i, z_i+1}|= 1}.

A polytope P with n ≥ 8 vertices is totally-sewn if there exist subpolytopes P7, . . . , Pn of P with the property that |V(Pm)| =m and Pm+1 is obtained from Pm by sewing. Since P7 is cyclic (cf. [12]), we may label its vertices such that P7= [x1, x2, . . . , x7] and the vertices satisfy Gale’s Evenness Condition in the order x1<· · ·< x7. Then it is easy to check that

U(P7) ={[x_i−1, xi]|i= 1, . . . ,7 andx0=x7}.

Assume that P_m+1 = [P_m, x_m+1] and x_m+1 is sewn through F(E_m, F_m, P_m) for 8≤m≤n. ThenP = [x₁, x₂, . . . , x_n] and the sewing order determines an ordering on the vertices of P by x1 < x2 <· · · < xn. Note that there may be more than one sequence of sewings that produce the same polytopeP fromP7. Once we fix a sequence of sewings, then we also fix the corresponding ordering of the vertices of P. If 1≤i < j≤n, then we may say that the vertexxj is after the vertexxi with respect to this ordering.

It is a great advantage of the sewing process that the universal edges and the facial structure of the new polytope can be completely characterized. The universal edges and facets of Pm+1 constructed in the sewing process are described by the following statement.

Lemma 2.2 (cf. [6]). Let P_m+1= [P_m, x_m+1]andx_m+1 be sewn through F(E_m, F_m, P_m)withE_m= [x_a, x_b]. Then

z₁ z_m−2 z₃ z_m−3

x_m+1

Pm/[xa, xb]

z₂

z₂

z₁ x_b z_m−2

z_m−3 z₂ z₁

xa z_m−2 z_m−3

P_m+1/[x_a, x_m+1] Pm+1/[xb, xm+1]

z₃ z₃

Figure 1.

• {[xa, xm+1],[xm+1, xb]} ⊂ U(Pm+1),

• any F∈ F(Pm+1)\ F(Pm)contains[xa, xm+1]or[xb, xm+1], and

• there is a labeling z1, z2, . . . , z_m−2 ofV(Pm)\ {xa, xb} so that F(E_m, P_m) ={[E_m, z_i, z_i+1]|i= 1, . . . , m−2andz_m−1=z₁}, and if Fm= [xa, xb, z1, z_m−2], then

F(Pm+1)\ F(Pm) ={[xa, xb, xm+1, z1],[xa, xb, xm+1, zm−2],

[xa, xm+1, zi, zi+1],[xb, xm+1, zi, zi+1]|i= 1, . . . , m−3}. We will frequently use in our arguments the representations of the quotient polytopes depicted in Figure 1. The first drawing describes the location of (the projection of) xm+1 in P/[xa, xb], and the second and third figures indicate the locations ofxb andxa in P/[xa, xm+1] and P/[xb, xm+1], respectively.

The most important ingredient in our proof is the concept of type of a vertex and a universal edge. This notion was introduced in [5]; below we recall the precise definition.

Let 6 ≤ k < v ≤ n, and let xv be sewn through F(Ev−1, Fv−1, Pv−1) with Ev−1∈ U(Pv−1). Then eitherEv−1 ∈ U(Pk) or Ev−1= [xt, xu] withxt< xu and xu > xk. In the latter case xu is sewn through F(Eu−1, Fu−1, Pu−1) and either E_u−1 ∈ U(P_k) orE_u−1= [x_r, x_s] withx_r < x_s and x_s > x_k. Iterating the above argument, we arrive at the conclusion that the vertexx_v originates from a unique universal edgeE∈ U(P_k) through a sequence of vertices that are sewn afterx_k and before x_v. Let U(P_k) ={E¹, . . . , E^l}. ThenE =E^α for some 1≤α≤l and we say that x_v (E_v−1) is a vertex (universal edge) of type α with respect to P_k. We note here that throughout the paper we use lower-case Greek letters to denote the vertex and universal edge types inP. We also recall from [5] the notations

V_k^α={xi∈ {xk+1, . . . , xn} |xi is typeαwith respect to Pk}, and

V_k^α(m) =V_k^α∩ {x_m, . . . , x_n} form > k.

In order to explain the importance of vertex (end edge) types, we used in [5] the similitude of sewingncoloured buttons on shirt. The buttons are the vertices and

the colours are the vertex types ofP. Once all the buttons are sewn in the order x7, . . . , xn, it turns out that the groupings of the colours are more important than the actual order of sewing.

Lemma 2.3 (Deletion process). Let 7 ≤ s≤ n and [x_p, x_q, x_r, x_s] ∈ F(P_s) with E_s−1= [xr, xt]andxt6∈ {xp, xq, xr, xs}. Then[xp, xq, xr, xt]∈(F(Pu)∩F(P_s−1))\ F(P)withu= max{p, q, r, t}.

Proof. The assertion is a direct consequence of Lemmas 2.1 and 2.2.

Note that the deletion process can be iterated if the facet [x_p, x_q, x_r, x_t] does not contain the sewing edgeE_u−1. In subsequent arguments we will iterate the deletion process in order to obtain facets at intermediate steps of the sewing process.

3. General properties of totally-sewn neighbourly 4-polytopes Lemma 3.1. Let 6 ≤m < s < w ≤n,x_s ∈V_m^α, and let x_w ∈V_m^β with E_w−1 ∈ U(P_s). Then E_w−1∈ U(P_s−1)andE_w−1∩E_s−1⊆E^α∩E^β.

Proof. Clearly, E_w−1 =E^β or E_w−1 is a β type edge with respect toPm. Since xs ∈ V_m^α, no new β type universal edge is constructed when xs is sewn. Hence, E_w−1∈ U(Ps) yields thatE_w−1∈ U(P_s−1).

Let xb be a common vertex of E_s−1 and E_w−1. If xb 6∈ Pm, then E_s−1 6= E^α which yields that xb ∈ V_m^α, and Ew−1 6= E^β which yields that xb ∈ V_m^β. This contradicts the fact thatV_m^α∩V_m^β=∅. Thus,xb∈Pm andxb∈E^α∩E^β. Lemma 3.2. Let 6 ≤k < m < u < w ≤ n, xm ∈ V_k^α, xu ∈ V_k^β, xw ∈V_k^δ and {xm, xu, xw} ⊂ F ∈ F(Pw), α6= β 6=δ 6=α. Then in U(Pk), E^β∩E^δ 6= ∅ and E^α∩(E^β∪E^δ)6=∅.

Proof. FromF ∈ F(P_w) and Lemma 2.2 it follows that there exists a vertexx_t∈F such that [x_t, x_w]∈ U(P_w),x_t∈E_w−1∈ U(P_w−1). Let E_w−1 = [x_s, x_t]. E_w−1= E^δ or E_w−1 is a δ type edge with respect toP_k. {x_s, x_t} ∩ {x_m, x_u} =∅ implies thatF = [x_m, x_u, x_t, x_w].

Application of the deletion process (cf. Lemma 2.3) to the facet F yields that e

F = [xm, xu, xs, xt] ∈ F(Pj) forj = max{m, u, s, t}. If xj ∈V_k^δ, then we iterate this deletion process. Hence, we may assume that, after a necessary number of iterations, Fe ∈ F(Pu). Then [˜x, xu] ∈ U(Pu) is a β type edge for some ˜x ∈ {x_m, x_s, x_t}. SinceV_k^α∩V_k^β=∅, it follows that ˜x6=x_m, and we may assume that

˜

x=x_s. Now, by Lemma 3.1, x_s∈E^β∩E^δ.

Since Fe = [xm, xt, xs, xu] ∈ F(Pu) and [xs, xu] ∈ U(Pu), there exists a ver- tex xr such that E_u−1 = [xr, xs]. Then Fb = [xm, xt, xs, xr] ∈ F(Pj) with j = max{m, t, r}. Now, we may assume that, after a necessary number of iterations of the deletion process,Fb∈ F(Pm). Then [ˆx, xm]∈ U(Pm) for some ˆx∈ {xt, xs, xr}. Since{xs}=E^β∩E^δ, it follows that ˆx6=xs. If ˆx=xr, then{xr}=E^α∩E^β, and if ˆx=xt, then{xt}=E^α∩E^δ by Lemma 3.1 In summary, ifF ∈ F(P) is a facet that satisfies the conditions of Lemma 3.2, then the universal edgesE^α, E^β and E^δ form a path of length three in U(P_k) in one of the following ways:

◦ ^Eα ◦ ^Eβ ◦ ^Eδ ◦ or ◦ ^Eβ ◦ ^Eδ ◦ ^Eα ◦.

Lemma 3.3. Let 6 ≤m < v < n, xm+1 ∈V_m^α, xv+1 ∈ V_m^β with {xv+1} =V_m^β∩ {xm+1, . . . , xv+1} and assume the(Pv/Ev, zi)-configuration. Let1≤i < j≤v−2, {zi, zj} ⊂ V_m^α, zi = xr, zj = xt and [Ev, zi, zj] ∈ F(Pt). Then (with suitable labeling) {z_i+1, . . . , z_j−1} ⊂V_m^α.

Proof. We note that Em = E^α, Ev = E^β, Em∩Ev ⊂Pm and Ev ∈ U(Pw) for m≤w≤v. Clearly, we may assume thati+ 1< j. Then

[Ev, zi, zj]∈ F(Pt)\ F(Pv) and there is at < u≤vsuch that

[E_v, z_i, z_j]∈ F(P_u−1)\ F(P_u).

Thenx_u=z_k for somei < k < j,E_u−1⊂[E_v, z_i, z_j]6=F_u−1and eitherE_u−1=E_v orE_u−1= [z_i, z_j] or|E_u−1∩E_v|= 1 =|E_u−1∩[z_i, z_j]|. We note thatE_u−1=E_v yields thatx_u=x_v+1; a contradiction. IfE_u−1= [z_i, z_j] = [x_r, x_t], thenx_t∈V_m^α implies thatE_u−1is anαtype edge with respect toPm andzk=xu∈V_m^α.

If E_u−1 = [xe, xr] with xe ∈ Ev = E^β, then xr ∈ V_m^α implies that {xe} = E^α∩E^β. Furthermore,E_u−1 isαtype and againzk=xu∈V_m^α.

Since [Ev, zi, zj]∈ F(P_u−1)\ F(Pu) clearly implies that {[Ev, zi, zk],[Ev, zk, zj]} ⊂ F(Pu),

the assertion of the Lemma readily follows from iterations of the argument above

for{zi, zk} and{zk, zj}.

In summary, Lemma 3.3 states that the α type vertices of Pv+1 determine a connected arc in the polygonPv/Ev.

4. Separation in totally-sewn neighbourly 4-polytopes

In this section we develop the basic tools that will be used in the proof of the main theorem. Some of the following statements are quoted from [5], and some are new. The arguments of the proofs are based on the lemmas of the previous section.

The following statement is a direct consequence of Lemma 2.1.

Lemma 4.1. Let 6≤k < n, H ⊂R⁴ be a hyperplane spanned by the vertices of Pk andxbe a point of R⁴.

4.1.1 If H strictly separates x and an endpoint of E^λ ∈ U(Pk), thenH strictly separatesxandV_k^λ.

4.1.2 If H strictly separates x and xu ∈ V_k^λ, then H strictly separates x and V_k^λ(u).

Lemma 4.2 (cf. [5]). LetQbe a4-dimensional subpolytope of P andO be a point of(intP)∩∂Q. ThenO is strictly separated from anyF ∈ F(P)∩ F(Q)by one of at most three hyperplanes.

Lemma 4.3. Let 6 ≤ m < u, v ≤ n, xm+1 ∈ V_m^α, {xu, xv} ⊂ V_m^β and xc ∈ V(Pm)\E^αsuch thathE^α, xc, xm+1i ∩[xu, xv] =∅. ThenhE^α, xc, xgi ∩[xu, xv] =∅ for allxg∈V_m^α.

Proof. Letxg ∈ V_m^α such that [xm+1, xg] ∈ U(Pg), and suppose, on the contrary, thathE^α, x_c, x_gi ∩[x_u, x_v]6=∅. ThenhE^α, x_c, x_gistrictly separatesx_uandx_v, and we may assume thathE^α, x_c, x_vistrictly separatesx_m+1 andx_g.

LetS denote the open region ofR⁴bounded byhE^α, xc, xm+1iandhE^α, xc, xgi, that containsxv. We note that ifxi ∈S, thenhE^α, xc, xiistrictly separatesxm+1

andxg. In addition,xu6∈Sand by Lemma 4.1S∩V(P)⊂ {xg+1, . . . , xn}. Without loss of generality, we may assume thatS∩V_m^β ⊂ {xv, . . . , xn}.

LetE_v−1= [xr, xs]. Then{xr, xs} ⊂V_m^β∪ V(E^β), and both [xr, xv] and [xs, xv] are universal edges ofPv and thus neitherhE^α, xc, xm+1inorhE^α, xc, xgistrictly separatesx_vfromx_rorx_s. Since{x_r, x_s} ⊂cl(S) and cl(S)∩V_m^β=S∩V_m^β, it follows from {x_r, x_s} ⊂ {x_r, . . . , x_v−1} that {x_r, x_s} ∩V_m^β =∅ and E_v−1= [x_r, x_s] =E^β and thusx_v is the firstβ type vertex with respect toP_m. Therefore,x_u> x_v and xv∈Syield thatxu∈S. This is a contradiction, and hencehE^α, xc, xgi∩[xu, xv] =

∅.

The statement of the lemma follows from iterations of the above argument.

The following lemma is the cornerstone of our argument.

Lemma 4.4. Let 6 ≤ k ≤ m < n, O ∈ intP_m, x_m+1 ∈ V_k^α and {x_m+1} = V_k^α∩ {xk+1, . . . , xm+1}. Let Fe∈ F(P)such that Fe∩V_k^α(m+ 1)6=∅. Then O is separated from any Fe by one of at most five hyperplanes spanned by vertices ofP. Proof. Note that V_k^α(m+ 1) = V_k^α. Let E_m = [x_a, x_b] = E^α, Q = [V_k^α], and assume the (P_m/E_m, y_i)-configuration. By Lemma 4.1, O is (strictly) separated fromQby each hyperplanehFˆifor ˆF ∈ F(Em, Pm)\{Fm}, and so there are vertices xg and xh of Q such that O is separated from any ˆF by one of the hyperplanes H1=hEm, yl, xgi,H2=hEm, yl, yl+1i,H3=hEm, yl+1, xhifor some 1≤l≤m−3, see Figures 2 and 3. The location ofOin intPmdetermines the choice ofl.

x_m+1

Q= [V_k^α]

H₁ H₃

y₁ y_m−2

y_l y_l+1

H₂

xg

x_h

O

Figure 2. Pm/Em

We examine first the facets Fe via the Deletion (process) and then consider the separation ofFe fromO based upon the location ofO.

We note that if E^λ ∈ U(Pk)∩ U(Pm), then V_k^λ = V_k^λ(m+ 1) = V_m^λ, and if E^λ ∈ U(P_k)\ U(P_m), andE^θ, . . . , E^ψ are the λ types in U(P_m) with respect to

Pk, then V_k^λ(m+ 1) = V_m^θ ∪ · · · ∪V_m^ψ. Thus, Fe∩V_m^α 6= ∅, and Fe contains at most two other types of vertices with respect to Pmby Lemmas 3.1 and 3.2. Let X ={xm+1, . . . , xn}andY ={y1, . . . , ym−2}.

i) LetFe∩X ⊂V_m^α. Then Fe∩Y ⊆ {yj, yj+1}for some1≤j≤m−3.

Proof of i). Let|Fe∩Y| ≥2,xu∈Fe∩V_m^α⊆ {xm+1, . . . , xu}andE_u−1= [xr, xs].

Since xu ∈ V_m^α and E^α ∩Y = ∅, it follows that E_u−1 ∩Y = ∅, E_u−1 6⊂ Fe,

|Fe∩Y|= 2 andxr orxsis inFe∩(V_m^α∪ {xa, xb}). Now, Deletion yields that there is anF⁰ ∈ F(P_m+1) such that F⁰∩Y =Fe∩Y and x_m+1 ∈ F⁰. By Lemma 2.2,

F⁰∩Y ⊆ {y_j, y_j+1}for some 1≤j≤m−3.

ii) Let Fe∩X ⊂ V_m^α∪V_m^λ and α 6= λ. Then either E^λ = [y_j, y_j+1] for some 1≤j ≤m−3andFe∩Y ⊂E^λ orE^λ= [E^λ∩E^α, y]for some y∈ {y1, y_m−2}and

e

F∩Y is contained in{y1, y2} or{ym−3, ym−2}.

Proof of ii). Since E^λ ∈ U(P_m)∩ U(P_m+1), we have that either E^λ∩E^α =∅, [E^λ, E^α] ∈ F(Pm) (cf. Theorem 3.4, [12]) and E^λ = [yj, yj+1] for some 1 ≤j ≤ m−3 orE^λ∩E^α6=∅andE^λ= [x, y] for some x∈ {xa, xb}andy∈ {y1, ym−2}.

Letx_p ∈Fe∩V_m^α ⊆ {x_m+1, . . . , x_p} and x_v ∈Fe∩V_m^λ ⊆ {x_m+1, . . . , x_v}. Since E_p−1∩Y ⊆E^α∩Y =∅andE_v−1∩Y ⊆E^λ∩Y, we may assume that

[E_p−1, xp, xv]6=Fe6= [E_v−1, xp, xv].

Then E_p−1 6⊂ Fe and E_v−1 6⊂ Fe, and it follows from Deletion that Fe ∩Y ⊆ (E_p−1∪E_v−1)∩Y in case E^α∩E^λ=∅.

LetE^α∩E^λ 6=∅ andFe = [xp, xv,x, ye i] withyi 6∈E^λ. By Deletion, we obtain that [x_m+1, E^α∩E^λ, x_w+1, y_i] ∈ F(P_w+1) with E_w = E^λ and [x_m+1, E^λ, y_i] ∈ F(Pm+1)∩F(Pw). Hence Lemma 2.2 yields thatyi=y2in caseE^λ= [E^λ∩E^α, y1] andyi =y_m−3in case that E^λ= [E^α∩E^λ, y_m−2].

iii) Let Fe∩V_m^δ 6= ∅ 6= Fe∩V_m^η and α 6= δ 6= η 6= α. Then E^δ ∩E^η 6= ∅ and E^α∩(E^δ∪E^η)6=∅.

Proof of iii). Let Fe = [xp, xq, xr,ex] with xp ∈ V_m^α, xq ∈ V_m^δ and xr ∈ V_m^η. It is clear that none ofE_p−1, E_q−1 andE_r−1 is contained inF, so by Lemma 3.2,e

{ex} ∈ {E^α∩E^δ, E^α∩E^η, E^δ∩E^η}

and we need only to verify thatE^δ∩E^η 6=∅. Letxu+1 ∈V_m^δ withEu =E^δ, and xt+1∈V_m^η withEt=E^η.

If {ex} =E^α∩E^η =Em∩Et then q = min{p, q, r} by Lemma 3.2. NowFe = [x_p,x, xe _r, x_q] and Deletion yields that [x_m+1,x, xe _t+1, x_q]∈ F(P_t+1), [x_m+1, E_t, x_q]∈ F(Pt)∩ F(Pq) andE_q−1∩Et6=∅. By Lemma 3.1,E_q−1∩Et=E^δ∩E^η.

If{ex}=E^α∩E^δ =E_m∩E_uthenr= min{p, q, r}, [x_m+1,ex, x_u+1, x_r]∈ F(P_u+1), [x_m+1, E_u, x_r]∈ F(P_r)∩ F(P_u) and∅ 6=E_r−1∩E_u=E^η∩E^δ. iv) Let Fe = [xp, xq, xr,ex] with xp ∈ V_m^α, xq ∈ V_m^δ, xr ∈ V_m^η, xu+1 ∈ V_m^δ with E_u=E^δ= [x_a, y₁], andx_t+1∈V_m^η with E_t=E^η= [y₁, y₂]. Then

• xe=xa,[xm+1, Eu, xr]∈ F(Pr)∩ F(Pu)andt < u, or

• xe=y₁, and either [x_t+1, E_u, x_p]∈ F(P_u)∩ F(P_t+1)andt < u, or[x_u+1, E_t, x_p]∈ F(P_t)∩ F(P_u+1)andu < t.

Proof of iv). From the Proof of iii), we have that xe∈ {xa, y1} and that we need only to verify the consequences ofex=y1.

Let Fe = [xq, y1, xr, xp]. Then p= min{p, q, r} and it follows by Deletion that F⁰= [x_u+1, y₁, x_t+1, x_p]∈ F(P_u+1)∪ F(P_t+1). The assertions are now immediate.

We now consider the separation ofO fromFe. We note that the location ofO is as indicated on Figure 2 or (due to symmetry) Figure 3. From i), ii), iii) and iv), we can determine what types of verticesFe may possess.

x_m+1

Q= [V_k^α]

y₁ y_m−2

y₂

x_g x_h

O

H₃ H₁

H₂

H₂⁺

Figure 3. P_m/E_m

Case 1. For some2≤l < m/2,Ois separated from anyFbas indicated in Figure 2.

If [yl, yl+1]∈ U(Pm), then we denote it by E^β, and ifV_m^β6=∅, then letxq ∈V_m^β with E_q−1 =E^β. From i), ii), iii), and Lemma 4.1, we obtain thatH₁, H₂ or H₃ separateO from anyFe such thatFe∩V_m^β=∅.

LetFe∩V_m^β6=∅. ThenE^β∩ {y₁, y_m−2}=∅and iii) yield thatFe∩X⊂V_m^α∪V_m^β, and ii) yields thatFe∩Y ⊆ {y_l, y_l+1}. Since Lemma 4.1 implies that H₂ separates O and Fe in the case H2 separates O and xq, we may assume that H2 does not separate O and xq. Then with a (P_q−1/E_q−1, zi)-configuration (cf. Figure 4), we have that H2 =hEm, yl, yl+1i= hxa, xb, E_q−1i is not a supporting hyperplane of P_q−1, and

z1≤zr=xa< xb=zs≤z_q−3 for some 1≤r < r+ 1< s≤q−3. Now, Lemma 3.3 yields

V_m^α∩ {z1, . . . , z_q−3} ⊂ {z1, . . . , zr} ∪ {zs, . . . , z_q−3},

and it follows from Lemma 4.1 and Fe∩Y ⊂Eq−1 that O is separated from any suchFebyH4=hE_q−1, xa, xg⁰iorH5=hE_q−1, xb, xh⁰ifor suitably chosenxg⁰, and x_h⁰ in V_m^β.

In summary,O is separated from anyFeby one ofH1, H2, H3, H4 andH5. Case 2. O is separated from any Fb as depicted in Figure 3.

If [xa, y1] ∈ U(Pm) then let E^δ = [xa, y1], and if V_m^δ 6=∅ then let xu+1 ∈ V_m^δ with Eu =E^δ. If [y1, y2]∈ U(Pm) then let E^η = [y1, y2], and ifV_m^η 6=∅ then let xt+1∈V_m^η withEt=E^η. LetH₂⁺ denote the open half-space determined by

H2=hEm, y1, y2i=hxa, xb, y1, y2i

that containsO. As in Case 1, we obtain from i), ii), iii) and Lemma 4.1 that

• Ois separated from anyFebyH2orH3in the case thatH₂⁺∩{xu+1, xt+1}=

∅, or

• O is separated from any Fe by one of H2, H3 and suitable determinedH4

andH5 in the case that|H₂⁺∩ {xu+1, xt+1}|= 1 andV_m^δ =∅ orV_m^η =∅. Let |H₂⁺∩ {x_u+1, x_t+1}| ≥ 1 and V_m^δ 6= ∅ 6= V_m^η. Then H₂ = hE_u, x_b, y₂i = hE_t, x_a, x_biand eitherm < u < torm < t < u.

x_q

[V_m^β]

z_r=x_a

x_b=z_s

H₄ H₅

z₁ z_q−3

O

x_g0

x_h0

H₂

Figure 4. Pq−1/Eq−1

Letm < u < tand refer to Figures 5 and 6 for configurations (Pu/Eu, zi) and (Pt/Et, wj). SinceEt= [y1, y2]∈ U(Pu)∩ U(Pu+1) andEt∩Eu ={y1}, it follows thaty₂∈ {z₁, z_u−2}and we may assume thaty₂=z₁. Thus, there is noF∈ F(P) such thatF∩V_m^δ 6=∅ and{y₂, z_j} ⊂F for somei≤j ≤u−2. For Figure 6, we note that{x_a, x_b}={w_i, w_j} for some 1≤i < i+ 1< j≤t−2.

Finally, iv) and u < t yield that if Fe∩V_m^δ 6= ∅ 6= Fe∩V_m^η for some Fe then [x_u+1, y₁, y₂,Fe∩V_m^α]∈ F(P_t) by Deletion. This means thatH₂ does not separate x_u+1and Fe∩V_m^α. Thus, by Lemma 4.1, x_m+16∈H₂⁺ implies thatx_u+16∈H₂⁺.

We note that by i), ii) and Lemma 4.1, O is separated from any Fe such that e

F∩(V_m^δ∪V_m^η) =∅ byH2 orH3. We now considerFe such thatFe∩(V_m^δ ∪V_m^η)6=∅. Letx_u+16∈H₂⁺. Thenx_t+1∈H₂⁺,V_m^α∩ {z₁, . . . , z_u−2} ⊂ {z₁, . . . , z_i}and

(V_m^α∪V_m^δ)∩ {w₁, . . . , w_t−2} ⊂ {w₁, . . . , w_i} ∪ {w_j, . . . , w_t−2}.

x_u+1 [V_m^δ]

z_i=x_b

z₁=y₂ z_u−2

H₂ x_h0

H₆

Figure 5. Pu/Eu

x_t+1 [V_m^η]

H₂

xa x_b

w₁ w_t−2

H₄ H₅

Figure 6. Pt/Et

Thus, O is separated from any Fe such that Fe∩X ⊂ V_m^α∪V_m^δ byH2, andO is separated from anyFe such thatFe∩X ⊂V_m^α∪V_m^δ ∪V_m^η andFe∩V_m^η 6=∅byH4 or H₅.

Letxu+1∈H₂⁺. ThenFe∩V_m^δ =∅orFe∩V_m^η =∅for anyFeby the preceding, and V_m^α∩ {z1, . . . , z_u−2} ⊂ {zi, . . . , z_u−2}. LetFe∩V_m^δ 6=∅. Then as already noted, we have thaty26∈Fe andFe∩X⊂V_m^α∪V_m^δ. Thus, it follows from ii) and Lemma 4.1 thatOis separated from any suchFebyH₆. LetFe∩V_m^η 6=∅. ThenFe∩X ⊂V_m^α∪V_m^η

by iii), and either xt+16∈H₂⁺ and H2 separatesO from any suchFe orxt+1∈H₂⁺ andH4 orH5 separateOfrom any such F.e

In summary,O is separated from anyFeby one ofH2,H3,H4,H5andH6. Finally, letm < t < u. Then we may consider Figure 5 to representPt/Et, and Figure 6 to representP_u/E_u. From iv), we obtain that ifFe∩V_m^δ 6=∅ 6=Fe∩V_m^η for some Fe thenx_t+1 6∈H₂⁺. We now argue as above and obtain that O is separated from anyFe by one of analogously labeledH₂, H₃, H₄, H₅ andH₆. In the following Corollaries: we assume the hypotheses and the notation of Lemma 4.4, and determine locations of O that do not require five separating hy- perplanes determined by vertices ofP.

Corollary 4.5. Let 1 ≤ i < j < p ≤ m−3, and O be contained in either the open region bounded by hE_m, y_i, y_ji, hE_m, y_i, y_pi and hE_m, y_j, y_pi or the relatively open region inhE_m, y_i, y_pi bounded by hE_m, y_iiandhE_m, y_pi, andx∈ {x_a, x_b}. If {[yi, x],[yi, yj]} 6⊂ U(Pm)then O is separated from any Fe by one of at most four hyperplanes. Moreover, if 1 < i,[y_i, y_j]6∈ U(P_m) and[y_j, y_p] 6∈ U(P_m) then three hyperplanes suffice.

Proof. Let i < 1, [yi, yj] 6∈ U(Pm) and [yj, yp] 6∈ U(Pm). Then O is separated from any Fe by one of H₆, H₆⁰ and hE_m, y_i, y_ji or one of H₇, H₇⁰ and hE_m, y_j, y_pi; cf. Figure 7. In case [y_i, y_j] or [y_j, y_p] is in U(P_m), we replace hE_m, y_i, y_ji or hE_m, y_j, y_piby two hyperplanes.

x_m+1 [V_k^α]

y_i

y_j

yp

y₁ y_m−2

H₇ H₆⁰

H₆ H₇⁰

Figure 7. P_m/E_m= [x_a, x_b]

Ifi= 1, and E^δ = [E^δ∩E^α, y1]∈ U(Pm) with V_m^δ 6=∅then we replace H6 by

two hyperplanes.

A similar argument yields

Corollary 4.6. Let1≤i < i+ 2< p≤m−3,Obe contained in the relatively open region in hEm, yi, ypi bounded by hEm, yii and hEm, ypi, and x∈ {xa, xb}. Then O is separated from anyFe by one of at most four hyperplanes. Moreover, if either 1< iori= 1 and[y1, x]6∈ U(Pm) then three hyperplanes suffice.

x_m+1 [V_k^β]

y₁=y_i y_m−2

yp

Figure 8. Pm/Em

LetU⁰(P6) ={[x6, x1]} ∪ {[xi, xi+1]|i= 1, . . . ,5}. We note thatU⁰(P6)⊂ U(P6) and that any x_w ∈ {x₇, . . . , x_n} is an η type with respect to P₆ for some E^η ∈ U⁰(P₆), and specifically, we start the sewing process withE₆= [x₆, x₁]. LetV^η= V₆^η and denote the elements ofU⁰(P₆) as indicated below.

U⁰(P₆) :_x◦

6

α _x◦

1

β _x◦

2

δ _x◦

3

θ _x◦

4

µ _x◦

5

λ _x◦

6

We letE^β=E_r−1in case V^β 6=∅, and introduce similarlyE_s−1=E^δ,E_t−1= E^θ,E_u−1=E^µ, andE_v−1=E^λ.

Finally, let H₁⁰ = hx1, x6, x2, x3i, H₂⁰ = hx1, x6, x3, x4i, H₃⁰ = hx1, x6, x4, x5i, H₄⁰ =hx1, x2, x3, x4i,H₅⁰ =hx1, x2, x4, x5i,H₆⁰ =hx1, x2, x5, x6i,H₇⁰ =hx2, x3, x4, x5i, H₈⁰ = hx2, x3, x5, x6i, H₉⁰ = hx3, x4, x5, x6i, and note that F(P6) = {H_i⁰∩P|i = 1, . . . ,9}.

SinceE^α= [x₁, x₆] and F₆= [x₁, x₆, x₂, x₅], it follows that H_i⁰∩P₆6∈ F(P) for i= 1,2,3. It is clear that ifV^η 6=∅ for someη6=α, then|F(P₆)∩ F(P)| ≤4. It is straightforward but tedious to check the following:

Remark 1. If there arek≥2types of vertices with respect to P6, then

|F(P6)∩ F(P)| ≤6−k.

We consider{x7, . . . , xn}=V^α∪V^β∪V^δ∪V^θ∪V^µ∪V^λ, the vertex type that is sewn last and facets ofP that contain a last sewn type vertex. Then we consider the vertex type that is sewn second last and the facets ofP that contain a vertex of second last sewn type but not a vertex of last sewn type. We reiterate this process

as often as necessary. Lemma 4.7 states that if O ∈ intP6, then O is separated from any such class of facets by one of at most three hyperplanes.

Lemma 4.7. Let O ∈ intP₆, V^η 6=∅ andE_w−1 =E^η ∈ U⁰(P₆). Let Fe ∈ F(P) such thatFe∩V^η 6=∅andFe∩V^τ =∅for allE^τ∈ U⁰(P₆)such thatE^τ =E_z−1and xw< xz. ThenOis separated from any suchFeby one of three hyperplanes spanned by the vertices ofP; moreover, at least one of the three separating hyperplanes may be chosen from {H_i⁰|i= 1, . . . ,9}=H⁰.

Proof. LetE_w−1 = [xa, xb] and assume the (P_w−1/E_w−1, yi)-configuration. Then P6= [xa, xb, yi₁, yi₂, yi₃, yi₄] with 1≤i1< i2< i3< i4≤w−3; cf. Figure 9.

x_w [V^η]

y_i2

y_i3

y_i4

y₁ y_i1

y_w−3 x_h0 x_g0

Figure 9. P_w−1/E_w−1

We use the same notation as in the proof of Lemma 4.4 fork= 6 andm=w−1.

We recall that three hyperplanes do not suffice to separateO ∈intP6 ⊆intP_w−1 from any suchFeonly ifFe∩V_w−1^ψ 6=∅withE^ψ∈ U(Pw−1) and eitherE^ψ= [yl, yl+1] orE^ψ= [xa, y1] (for example); cf. Figures 2 and 3.

If 1 < i1 < i1+ 1 < i2 < i2+ 1 < i3 < i3+ 1 < i4 < w−3, then any such E^ψ is separated from O by one of H1, H2 and H3. In this case 1 < ij ≤ l <

l+ 1≤ij+1 for some 1≤j ≤3. Then there existxg⁰, xh⁰ ∈V^η such thatH1, H2

andH3 can be replaced by the hyperplaneshE_w−1, yi_j, xg⁰i,hE_w−1, yi_j, yi_j+1iand hE_w−1, yi_j+1, xh⁰i, respectively. We note that the hyperplane hE_w−1, yi_j, yi_j+1i is listed in H⁰. To simplify notation we rename H1 = hEw−1, yi_j, xg⁰i and H3 = hEw−1, yi_j+1, xh⁰i.

IfE^ψ = [yl, yl+1] = [yi_j, yi_j+1] andyl+1 =yi₄ or ifE^ψ = [xa, y1] and y1 =yi₁, thenE^ψ∈ U⁰(P₆),E^ψ=E_z−1for some x_z> x_w, and hence,Fe∩V^ψ =∅. Remark 2. We note that in the case wheni1= 1andi4=w−3, the hyperplanes hE_w−1, y_i1, y_i2i,hE_w−1, y_i2, y_i3iandhE_w−1, y_i3, y_i4ihave the same separation prop- erties as H₁,hE_w−1, y_ij, y_ij+1iandH₃ regardless of the position ofO inP₆. Thus,