Bounding the Numbers of Faces of Polytope Pairs and Simple Polyhedra

Annals of Discrete Mathematics 20 (1984) 215-232 North-Holland

BOUNDING THE NUMBERS OF FACES OF POLYTOPE PAIRS AND SIMPLE POLYHEDRA Carl W. LEE* ZBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598, U.S.A.

Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506. U.S. A.

Let P be a simplicial d-polytope with Y vertices and m(P) be the simplicial (d - I)-complex associated with the boundary of P. Suppose, for a given face F of P, that we know the numbers of faces of various dimensions of lkz(plF.Then we are able to determine upper and lower bounds for the possible numbers of faces of all dimensions of P and of P(P)\F. As a consequence, we can bound the numbers of faces of a simple d-polyhedron P if the numbers of bounded and unbounded facets of P and the dimension of the recession cone of P are specified.

1. Introduction Klee [7] in 1966 proved that every simple d-polyhedron P with v facets has at least v - d + 1 vertices. Griinbaum [6, Section 10.21 speculated whether this result might be improved upon if one specified both the number of bounded and of unbounded facets of P. In 1974 Klee 181 approached problems of this form from the point of view of pairs of simple polytopes while investigating the efficiency of a proposed algorithm to enumerate the vertices of a simple polytope defined by linear inequalities. By taking advantage of the recently established McMullen’s conditions for the numbers of faces of simplicia1 polytopes, Billera and Lee [5,9] were able to strengthen Klee’s results. In this paper we offer some further extensions to the above results, and as a consequence will be able to provide (often tight) bounds on the numbers of faces of a simple unbounded polyhedron with recession cone of a specified dimension. In the next three sections we review some preliminary material that is largely (though not entirely) discussed also in [5]. The main results of this paper are presented in the remaining two sections. 2. Polyhedra and f-vectors

Let P be a d-polyhedron (convex d-dimensional polyhedron). Faces of P of dimension 0, 1, d - 2 and d - 1 will be called vertices, edges, ridges (or subfucers) * This research was supported, in part, by the National Science Foundation under grant MCS77-28392 and by a National Science Foundation Graduate Fellowship. 215

216

C.W.Lee

and facets of P, respectively. The set of vertices of P will be denoted V(P). We will assume that any polyhedron under consideration has nonempty vertex set. Recall that every unbounded d-polyhedron P can be decomposed as P = 0 + K, where Q is a d-polytope and K is a polyhedral cone, called the recession cone of P. A d-polyhedron P is simple if every vertex of P is contained in exactly d facets of P. A d-polytope P is simplicial if every face of P is a geometric simplex. Let 9;be the set of all simplicial d-polytopes. For every simplicial (respectively, simple) d-polytope P one can find a simple (respectively, simplicial) d-polytope P* that is dual to P in the sense that there is an inclusion-reversing bijection between the set of faces of P and the set of faces of P*. For integer 0 S j s d - 1, let f,(P) denote the number of j-faces (jdimensional faces) of P. The d-vector f ( P ) = ( f , ( P ) ,fl(P), . . . ,fd-l(P))is called the f-uecfor of P. Define f(P;) = {f(P): P E 93.When P E 9;we will also set f - , ( P ) = l ,and f,(P)=Oif j < -1 or j > d - 1 . 3. Simplicia1 complexes A simplicial complex A on the finite set V = V ( A ) is a nonempty collection of subsets of V with the property that { u } E A for all u E V and that F E A whenever F C G for some G E A. For F E A we say F is a face of A and the dimension of F, dim F, equals j if card F = j + 1. In this case we call F a j-face of A. The dimension of A, dim A, is defined to be max{dim F: F E A}. If dim A = d, we will refer to A as a simplicial d-complex. Analogously to polyhedra, for simplicial (d - 1)-complex A we define the uerrices, edges, ridges, facets and f-vector of A. Two simplicial complexes, A , and A2, are isomorphic, denoted A, = A2, if there is a bijection between V(A,) and V(Az) which induces a bijection between A , and A,. We will often write u I u 2 * -u*k for the set F = { u l , u 2 ,... , u k } and P or u I u 2 .. . u k for the power set of F. By 1 A 1 is meant the underlying topological space of the simplicial complex A. If / A I is a topological d-ball (respectively, d-sphere), we say A is a simplicial d -ball (respectively, simplicial d-sphere). For simplicial d-ball A, write aA for the simplicial (d - ltsphere associated with d 1 A 1. This complex is called the boundary of A and it is known that aA = u{F: F is a ridge of A contained in exactly one facet of A}. Let A be a simplicial complex. For any F E A , the link of F in A is the simplicial complex lkaF = {G E A : G n F = 0, G U F E A}. Furthermore, if Ff 0, the deletion of F from A is the simplicia1 complex A \F = (G EA: F g G } .

Bounding the numbers of faces of polytope pairs

217

For simplicial (d - 1)-complex A, define the polynomials

c h(A)t'"

d-1

f ( A ,t ) =

j=-l

and

The h-vector of A is the ( d + 1)-vector h ( A ) = ( h a ( A ) ,h l ( A ) ,. . . ,h , ( A ) ) determined by the polynomial relation

c hi( A ) t i . d

h ( A ,t ) =

i =O

We also set h, ( A ) = 0 if i < 0 or i > d. (McMullen and Shephard [ l l ] write g ! ? , ( A )instead of h i ( A ) , and use

c

d-1

f(A,t )=

j=-1

( - lY"fj (A)t'"

and

g'd'(A, t ) = (1 - t)"f ( A ,

-&)

2 gjd'(A)ti+'

= i=-l

to define the g:d'(A).)The hi ( A ) can be written explicitly as linear combinations of the h ( A ) by

2

d-j h i ( A ) = j =O ( - l ) i - i(d - i ) fj-l(A),

O S i Sd,

and f ( A ) can be recovered from h ( A ) by

or by j+l

h ( A ) = c ( d -dj -- il )

hi(A),

-1sjsd-1.

i=O

We also define gi( A ) = hi ( A )- L ( A ) for all integer i.

4. Polyhedral complexes Let P be a simplicial d-polytope. The simplicial (d - 1)-complex Z(P) associated with P is defined to be Z ( P )= { F C V ( P ) :conv F is a face of dP},

C.W.Lee

218

where convF denotes the convex hull of F and dP means the boundary of P.It will be natural sometimes to abuse notation and refer to an F E X ( P ) itself as a face of P, but it should always be clear from the context whether by F we mean a face of P or a face of Z(P).Because such a simplicial complex is a simplicial sphere, we will call it a polyhedral (d - 1)-sphere. We will write f(P,t ) , h(P, t ) , etc. for f ( X ( P ) ,t ) , h ( Z ( P ) ,t ) , etc. and call h ( P ) the h-uector of P. Define also h ( 9 : ) = { h ( P ) : P E P:}. Now suppose 1 S k C d are integers, P is a simplicial d-polytope and F is a (k - 1)-face of P. Then the simplicial complex A = Z(P)\F is a simplicial (d - 1)-ball, and will be called a polyhedral (d - l ) - b d . Let P* be a simple d-polytope that is dual to P and F* be the (d - k)-face of P* corresponding to E Define Q * = P* F* to be the unbounded d-polyhedron obtained from P* by applying a projective transformation [6, Section 1.1; 11, Section 1.21 that sends a supporting hyperplane defining F* onto the hyperplane at infinity. Then Q " will have a recession cone of dimension d - k + 1, and A is dual to Q * in the sense that there is an inclusion-reversing bijection between the faces of A and the nonempty faces of Q*. In fact, every unbounded, simple d-polyhedron is dual to Z ( P ) \ v for some simplicial d-polytope P with vertex u, and every unbounded, simple d-polyhedron with (d - k + 1)-dimensional recession cone is dual to 2 ( P ) \ F for some simplicial d-polytope P with (k - 1)-face F. If A is a polyhedral sphere or ball, we say that A is a polyhedral complex. We now summarize some operations that can be performed on these complexes. Let A , and Az be simplicial complexes on disjoint vertex sets. The join of A, and Az is the simplicial complex A, * Az = {FIU F2: Fl E Al, Fz E &). In this case h(Al A2,t ) = h(A,, t)h(A2, r ) . If A l and A2 are polyhedral complexes, then so is A , - A 2 [12, 51. Let A be a simplicial (d - 1)-complex and F be a facet of A. For ubf V(A), the stellar subdivision of F in A is the simplicial complex st(u,F)[A] = (A \F)U 6 . #-. In this case, h,(st(u,F)[A]) = hi(A) if i = 0 or i = d , and h,(st(v, F)[A]) = hi ( A ) + 1 if 1 s i S d - 1. If A is a polyhedral (d - 1)-sphere (respectively, polyhedral (d - 1)-ball), then st(u, F)[A] is a polyhedral (d - 1)sphere (respectively, polyhedral (d - 1)-ball) (12, 51. For A a simplicial (d - l)-complex, u E V(A) and u E V(A), the simplicia1 wedge of A on u is the simplicial complex w(u,v)[A] = { ~ , , , u } . ( A \ u ) U u v . I k a u . If A isa polyhedral (d -1)-sphere, then w(u,u)[A]is a polyhedra1 d-sphere, Ik,,,,wAp =A, and h,(w(u,u)[A])= h,-,(A)+ h,(A)- L I ( l k A u ) ,0 6 i 6 d + 1 [12, 51. By repeating the wedging operation, we can, for k 2 1, define the simpliciul k-wedge of a polyhedral (d-1)-sphere A on a vertex u of A, given u 1 , u 2 ..., , u k g V(A),bytaking Wk(uIU2*.*uk,u)[A]t o b e w(u,,u)[A]if k = 1 and to be w(uk, uk-l)[wk-'(ul. * uk-1, u)[A]J, if k 3 2. In this case,

-

-

210

Bounding the numbers of faces of polytope pairs W ‘(u1

and

. . uk, U ) is a polyhedral (d + k

- 1)-sphere, I k w * ( u , . . . u k , u ) l ~ ] U 1 *

.

*

uk

= A,

Two kinds of simplicial d-polytopes figure prominently in the history of f-vector problems. The first is C ( v ,d ) , the cyclic polytope of dimension d with v vertices [6, Section 4.7; 111. The Upper Bound Theorem [lo, 111 states that fi (P)S fi (C(v,d)), 0 S j S d - 1, for all d-polytopes P with v vertices, and its proof uses properties of the h-vector of the cyclic polytope. It is known that

where [ x ] denotes the greatest integer not exceeding x. In fact, the Upper Bound Theorem was proved by showing that h, (P)zs h, (C(v,d)), 1 6 i S d, for all simplicial d-polytopes P with v vertices. Adopting the convention that (-d) = 1 and otherwise (:) = O if a < b or b 1; and hi (C(v,1)) for v > 2 using the above formula, even though such polytopes do not exist. Consistent with this definition we may take

f-,(C(v,O))= 1 for Y > 1; and f-,(C(v, 1))= 1 and fo(C(v,1))= 2 for v > 2 . Finally, we define h, (C(v, d ) ) = 0 = f , (C(v,d)) for all i and j if d < 0. The other type of simplicial d-polytope of interest is P ( v , d ) , defined recursively as follows: P(d + 1,d) is any geometric d-dimensional simplex, and for v > d + 1, P ( v , d ) is obtained from P ( Y - 1,d ) by building a ‘pyramidal cap’ over any one of the facets of P ( v - 1, d). (Thus, the boundary complex of P(v,d ) is derived from the boundary complex of a geometric d-simplex by a sequence of v - d - 1 stellar subdivisions.) Such polytopes are usually called stacked polytopes. The simplicial d-polytopes of type P(v, d ) are not combinatorially equivalent, but they all have Y vertices (if d > 1) and the same numbers of j-faces, and the Lower Bound Theorem [l,21 asserts that fi (P) t;. ( P ( v ,d)), 0 =sj S d - 1, for all simplicial d-polytopes P with v vertices. The h-vector of P(v, d ) has a particularly simple form: i=Oori=d, hi ( P ( v , d ) ) =

v-d,

1SiSd-1,

C. W. Lee

220

and the f-vector of this polytope is

(v-d)(d-l)+2,

j=d-l.

Again, we formally define h(P(v,O))= (1) and f-,(P(v,O))= 1 if v > 1; h(P(v,1)) = (1, l), f-,(P(v, 1)) = 1 and f o ( P ( u ,1)) = 2 if v > 2; and hi(P(v, d))= O = f i ( P ( v , d ) ) for all i and j if d
k = ( l ' ) + ( ? - '1 )-+1 where ni > ni-l> . . - > n, 3 j

...+(?), 3 1, from

which is defined

).

k " ' = ( n; ++l1) + ( n,-, + 1 )+...+( n, + 1 j+l

We also define O"'=O for positive integer i. A (d +I)-vector of integers (ho,h , ,. .. ,hd) is called an 0-sequence (or M-vector) if ho = 1, h, 3 0, 1< i s d, and h,,, =sh Y , 1c i s d - 1. Our primary tool in the study of f-vectors and h-vectors of polyhedral complexes is the characterization of h ( 9 : ) given by

Theorem 1 (McMullen's Conditions). Let h = (ho,h,, . . . ,h d ) be a (d + 1)-vector of integers, go= ho, and gi = hi - h,-,, 1 S i S n = [ d / 2 ] . Then h E h ( 9 f ' )if and only if the following two conditions hold: (1) hi = h d - 8 , 0 s i 6 n (the Dehn-Sornrnemille equations); ( 2 ) (go,g , , . . . ,go) is an 0-sequence. In particular, we remark that (2) implies hoG h, S . . h-vector of a simplicial d-polytope must be unimodal.

5 h,,

and hence the

5. Polytope pairs

A polytope pair ( P , F ) of type (d, v, k,h), where 1 k s d < v and h E h(B:lVk), is a simplicial d-polytope P with Y vertices and a (k - 1)-face F E 2(P)such that h = h(Ikz(p,F). We remark that there exists a simplicial ( d - k)-polytope Q such that E(Q) is isomorphic to Ikz:(p,F(take Q to be the 'quotient polytope' PIF [ l l , Section 2.21); hence the necessity of h E h(B,d-k).

Bounding the numbers of faces of polytope pairs

12 1

Naturally, McMullen's conditions also hold for h(P). It is easy to see that fo(lkZcp,F)Sv - k and hence that h, S v - d. Let A = Z(P)\F. Then a simple calculation yields f, (P) = f, ( A ) + f;.-k ( l k ~ , ~ , F for ) all 1 and hi (P) = hi (A) + hi-k(lkI(p)F)for all i. Now let P* be a simple d-polytope dual to P, F* be the (d - k)-face of P * corresponding to F, and Q * = P* F*. Duality then yields

-

( P )= fd-j-l(P*),

0 sj

S

d - 1,

f,(A) = fd-j-l(Q*),

0 sj

S

d - 1,

fj

f,(aA)=f'd2ji-l(Q*), f , (lkz&)

= fd-k-j-i(F*),

OSjsd-2,

0S J s d

-k

-

1,

where fY'(Q*) denotes the number of unbounded j-faces of Q*. In particular, the number of facets of Q * is v - 1 if k = 1 and v if 2 S k S d, and the number of if k = 1 and fo(lkz(p,F)+ k if 2 S k s d. unbounded facets of Q * is fo(lkZcp,F) First, let us consider the easy cases of k = d - 1 and k = d. If F is any facet of any simplicial d-polytope P, then IkZ(p,F is a simplicial ( - 1)-sphere and h(Ik,,,,F) = (1). Similarly, if F is any ridge of any simplicial d-polytope P, then lkZ(p)Fis a simplicial 0-sphere and h(IkZ(p)F)X (1,l). McMullen's conditions therefore allow us to characterize completely { h ( P ) } and { h ( Z ( P ) \ F ) } for all polytope pairs (P,F) of type (d, v, k, h), when k = d - 1 or k = d. As a result of duality, the Upper Bound Theorem and the Lower Bound Theorem, we have Theorem 2. Let 3 S d < v. Assume that either k = 1 and r = d ;or else that k = 2 and r = d 1. As P ranges over all simple d-polyhedra with recession cone of dimension k, and v facets, r of which are unbounded, then

+

(i)

(ii)

minfi(P) =

maxf,(P) =

i i

fd-l(P(v, d)) - k, if j = 0, fd-Z(P(V,d))- 1, i f j = 1 and k = 2 , fd-,-l(P(v, d)), if k S j s d - 1;

fd--L(C(V,d))-k, if j =o, fd--2(c(v, d))- 1, if j = 1 and k = 2, fd-,-l(c(v, d)), if k S j S d - 1.

Moreover, for either value of k, there exist simple d-polyhedra P1 and Pz satisfying the above conditions such that f (PI)achieves all of the values in (i) and f(Pz)achieves all of the values in (ii). Now fix 3 s d < v, 1 s k s d -2and h E h(9~-')such that 1 s h l S v -d. As (P,F) ranges over all polytope pairs of type (d, v, k, h ) define A :(d, v, k, h ) = min hi(P),

Osisd,

222

C.W. Lee

A :(d, v, k, h ) = min h, (2(P)\ F),

0s i

~ : ( dv,, k, h ) = max h, ( P ) ,

Osisd,

pL(d,v,k,h)=maxh,(C(P)\F),

Osisd.

S

d,

Of course, the Dehn-Sommerville equations imply that A!(d,v, k, h ) = Afi-,(d, v, k, h ) ,

0 6 i s [d/2],

pj(d, v, k , h ) = pccl-,(d,v, k, h ) ,

0 s i s [d/2].

Our goal is to provide bounds on these values and to investigate the tightness of these bounds. In [5] we consider the case k = 1 and prove the following three results:

r

Theorem 3. Let 3 S d < v and h E h ( 9 ) f ' ) such that 1 S h, s v [ d / 2 ] and m = [ ( d - 1)/2]. Then

(i)

A l(d, v, 1, h ) =

(iii)

p f ( d ,v, 1, h ) =

Put n =

i =0,

v" - d - h l + h i ,

c

- d.

lsisn;

i =0,

1,

rdY-* )+ hi-,,

0 si s n;

Moreover, there exist polytope pairs ( P , ,u , ) and (P2, u2) of type (d, v, 1, h ) such that

hi ( P I )= A f (d, v, 1, h ),

0 s i G d,

hi (2(PI)\ul) = A:(d, v, 1, h ) ,

0S i

S

hi ( P z )= p ( d , v, 1, h ) ,

0S i

6 d,

hi ( 2 ( P 2 ) \v z ) = pS(d, u, 1, h ) ,

0S i

S

Corollary 1. Let 3 s d

Gr

d, d.

< u and put n = [d/2] and rn = [ ( d - 1)/2]. Then as P

Bounding the numbers of faces of polytope pairs

223

ranges over all simplicial d-polytopes with u vertices, one of which, v, is on exactly r edges, we have the following minima and maxima function (i)

hi (lkr(p,v)

(ii)

hi(P)

minimum

i

i =0,

r”- d + l ,

l ~ i ~ m ;

i =0,

u-d,

l ~ i ~ n ; i =0,

(iii)

i=l,

hi (2( P )\ u )

2 ~ i ~ d - 1 , i=d:

( v - r - l ) ( d - 2) + v - d,

(viii) h, ( P )

( v - d + i - 2 ) + ( r - d i -+1i - 1

I

(v-dlTi-2),

OX>

hi(s(P)\v)

(v d- i-- i2 ) ,

IB

-

,

O c i ~ n ;

GiGn,

n

+ 14 i s d - 2 ,

i=d-1,

-

j=d-1;

i=d;

C.W. Lee

224

(x)

(xi)

fi(Ik=,p)u)

fim

fi(C(r,d - 1)),

0 s j s d -2;

fi (C(v - 1, d )) + fi (C(r + 1, d )) - fi (C(r, d )), 0 s j

6 d - 1;

Moreover, there exist polytopes PI, P2and P3 of the above form with vertices 0 1 , ul and 0 3 , respectively, such that h, (Ikp(p,)uI), h, (Pl), h, (.Z(P,)\u,), f r ( I k 2 ( ~ , ~ 1fi(P1) ), and fi(Z(Pl)\ul) are the values given in (i) through (vi), respectively; h, (Ikx(p2)u2), h, (PZ), fr(Ikz(p,pZ) and f , (P2)are the values given in (vii), (viii), (x) and (xi), respectively; and h,( 2 ( P 3 ) \u3) and fi (.Z(P3)\u3) are the values given in (ix) and (xii), respectively. Corollary 2. Let 3 S d S r s v. A s P ranges ouer all simple d-polyhedra with v facets, exactly r of which are unbounded, then

[ ( u - r ) ( d -2)+

(ii)

maxh(P) =

v-d

+ 1,

fd-l(C(v,d ) ) + d - r - 1,

f i 2 ( C ( v , d ) ) + d- r , fd-j-l(c(v, d ) ) ,

j =0,

j = 0,

j=1, 2 s j s d - 1.

Moreover, there exist simple d-polyhedra P1 and P, satisfying the above conditions such that f(P1)achieves all of the values in (i) and f ( P 2 )achieves all of the values in (ii). We now turn to the polytope pairs with 2 s k s d - 2. We can determine the values of A j(d, v, k, h ) and h?(d,u, k, h ) , but at the present time must content ourselves with placing upper bounds on p ! ( d ,v, k, h ) and pLf(d, v, k, h ) in most cases. Theorem 4. Let 4 s d < v, 2 s k s d - 2 and h E h(CPt-5 such that hl v - d. Let n = ( d / 2 ] , m = [(d - k)/2] and p = [(d - k + 1)/2]. Then (with the conuention that h, = 0 if i < 0): (i)

h f ( d ,v, k, h ) =

v - d - hl+ h,,

[:-d-h,+hm,

i =0, lsiSm, m +lSiSn;

Bounding the numbers of faces of polytope pairs

(ii)

A ?(d,v, k, h ) =

(iii)

p :(d, v, k, h ) G

(iv)

pFLf(d,u, k, h ) c

-d

i =0, 1 S i S m, m + 1 S i G d - rn d - m Si s d - 1 , i=d;

h, - hi-k, - hl + h, - hi-k,

(v -d T i - 1 ) - (v -d

c

225

+ i -k -1 + hi-k,

i-k

u-i-k-1 d-i-k

+

hd-i-k

-

1,

0 siG n;

- hi-k,

n + 1 S i 6 d.

Moreover, there exists a polytope pair (P*,F*)of type (d, u, k, h ) such that

h i ( P * ) = A : ( d , ~k,, h ) ,

OGiSn,

hi(.Z(P*)\F*)= A f ( d , u, k, h ) ,

0 si

G

d.

Finally, the upper bounds in (iii) and (iv) are achievable (1) if O ~ i s or p d-psisd, (2) for all i if hi = ("-df'-'), 0 6 j C m, or (3) for all i if k S [(d + 1)/2] and hi = hr(d+l),2]-k, [ ( d + 1)/2] - k + 1 S =S m, in each case there being a polytope pair ( P * , F * )of type (d, v, k, h ) achieving these upper bounds simultaneously.

Remark. If k 3 n,then the inequalities in (iii) and (iv) say simply p f (d, v, k, h ) S ("-d:i+'), 0 s i S n ; p f ( d ,u, k, h ) c ( " - d : i - ' ) , OS i < n ; and p f ( d ,u, k, h ) s ("ik:')- hi-k, Iz + l < i s d . Proof. Note that the Dehn-Sommerville equations imply that h, (P) = hd-,( P ) , 0s i S d, and h, = hd-k-,, 0 S i s d - k, for every polytope pair of type (d, v, k, h ) .

Establishing the bounds. We will determine lower bounds for A f (d, u, k, h ) and upper bounds for pj(d, v, k, h ) by induction on k, from which we obtain bounds for A f ( d ,v, k, h ) and p f ( d ,v, k, h). First observe that we have equality in (i) and (ii) and we have equality with the upper bounds in (iii) and (iv) when we set k = 1 by Theorem 3 (using for (i) and (ii) that m = n if d is odd and h, = h, by the Dehn-Sommerville equations if d is even). So assume k 3 2 and (P, F) is a polytope pair of type (d, v, k, h). Let u be any vertex of P in F, take 0 to be a

C.W.Lee

226

vertex figure of P at u, i.e. Q = H n P for some hyperplane H that strictly separates u from the remaining vertices of P, and put V’ = V(F)\u. Then G = conv V’ corresponds to a (k - 2)-face of Q, which we will call G also. Note that Ikz,p,F= lkik,,,,,G= lkL,o,G. Thus (Q, G ) is a polytope pair of type (d - 1, v’, k - 1, h ) for some v ’ 6 v - 1, and (P, u ) is a polytope pair of type (4 v, 1, h ( 0 ) ) . and h, ( 0 )3 Now h, ( P )L A ( d , v, 1, h (a)), h i ( Q )= v’ - d + 1 A f(d - 1, v’,k - 1, h). So by Theorem 3 and induction:

h , ( P ) av - d - h , ( Q ) + h , ( Q ) ,

-I

v

-

d - (v‘- d

v-d -(v’-d

=[

l s i s n

+ 1) + (v’- d + 1) - hi + h,, + l ) + ( ~ ’ - d + l ) - h l + h,,

m

v - d - hi + h,, v - d - hi

+ h, ,

l s i s m

+ 1 s i s [(d - 1)/2]

l
Also, as a consequence of McMullen’s conditions, h,, (P)2 h, (P)5 v - d - h i + h,. Therefore, the values in (i) are lower bounds for the A:(d,v,k,h). Using the remarks in the beginning of this section, we have

n + l < i s d - m -1;

=v-d-h,,

d-rnsisd-1.

Therefore the values in (ii) are lower bounds for the A f ( d , v, k,h ) .

227

Bounding the numbers of faces of polytope pairs

For the upper bounds in (iii), h i ( P ) s p i ( d ,v , l , h ( Q ) ) and h i ( Q ) S p : ( d - 1, v', k - 1, h ) S p : ( d - 1 , v - 1,k - 1,h ) . (To justify this last inequality, we observe that if (P,F ) is any polytope pair of type (d, v, k, h ) , and P' is obtained from P by performing a stellar subdivision of some facet of P not containing F, then (P',F ) is a polytope pair of type (d, v + 1 , k, h ) , and hi (P')3 hi(P)for all i.) So by Theorem 3 and induction, for 0 S i s n we have hi(P)S(

u-d+i-2 u

-(u -

) + hi-l(Q)

-d +i -2)

+

- d ; i -1)-

+i -2)

-( u -d

(u

-d

(u

- d + i -k -1 i-k

i-1

+i -k -1 i-k

+ hi-*

+ hi-k.

Therefore we have the upper bounds in (5). The upper bounds in (iv) come immediately from those in (iii) using the relation h , ( Z ( P ) \ F ) = h , ( P ) - h , - k .

Achieving the bounds. Because the bounds were derived using the relation h, ( x ( P ) \ F ) = h, ( P )- h,-k,it is sufficient to find a polytope pair ( P * ,F * ) of type ( d , v, k, h ) such that h ( P * )achieves all of the bounds for h : ( d , v, k, h ) simultaneously, for then we may conclude that h ( Z ( P * ) \ F * ) achieves all of the bounds for A f(d, v, k, h ) simultaneously. By [5, Theorem 3.131 there exists a simplicia1 (d - k)-polytope P such that (i) h, ( P )= h,, 0 G i s d - k, and (ii) there is a vertex u of P such that h, (lkr(p,u)= h,, 0 S i S [(d - k - 1)/2]. Let ~ = w k ( ( u l ~ . . u k , u ) [ ~where ( P ) ] ,u I , u 2,..., u k E V ( P ) . Now put Z o = X and for integer j 3 1 let CJ= st(u,, G,-l)[Zl-,] for some v,E V(&) and some facet GI-,of ZJ-, not containing u l u 2 . * uk. Let P* be a simplicial d-polytope and put F* = conv(ul,uz,...,uk}. Using the facts such that 2 ( P * )= Ld-,,,, about stellar subdivisions and wedges it can be deduced that (P*,F*) is a polytope pair of type ( d , v, k, h ) achieving all of the bounds of (i), and hence also of (ii), simultaneously. To achieve the upper bounds of (iii) and (iv) for case (l), by [5, Theorem 3.131 there is a simplicia1 (d - k)-polytope P and a simplicial ( d - k 1)-polytope Q such that (i) h ( P ) = h ; (ii) Q has u - k + 1 vertices;

+

C. W. Lee

228

(iv) P is a vertex figure of Q at a vertex z E V ( 0 ) . Let 1: = w k - ' ( u , .. U k - 1 , z ) [ Z ( Q ) ] where , u I , .. . ,u k - 1 v(Q),and let P* be a simplicia1 d-polytope such that 2 = Z ( P * ) . Then Ik,r(p*lUt. . *

W - ~ Z= k k z ( p . , u ,

ut-,z

= Ikx(0lZ =Z(P).

Put F* = u I . . . u k - ' z . Then h(Ikrcp.,F*)= h ( P )= h. Also, P* has v vertices, so ( P * ,F*) is a polytope pair of type (d, v, k, h ) . For 0 S i S p , h,(P*)=

8

hj(Q)-

+I

,=i-

2

hj(P)

j=i-k+l

2

2 [ ( u - d +i j - 2 ) +

= j=i-k+l

- (v -

-d

T i - 1) - ( v - d

j=i-k+l

+i -k -1 i-k

h, hi-k.

Hence, h,( P * ) achieves the upper bounds in (iii) if 0 6 i S p , and h i ( 2 ( P * ) \ F * ) i d. achieves the upper bounds in (iv) if 0 s i < p or d - p Achieving the upper bounds of (iii) and (iv) in the case ( 2 ) is straightforward, taking P* to be C ( u , d ) with ordered vertex set { v l , v 2 , . . . , v v } , and F* = v 1 u 2 .* . vk. The construction of a polytope pair to achieve the upper bounds in the case (3) is more tedious than illuminating, so we will simply refer the reader 0 to [9, Section 5.31 for the details. Corollary 3. Lei 4 s d < r < v and 2 ~ k S d - 2 . Put n = [ d / 2 ] and m = [(d - k ) / 2 ] . As P ranges over all simplicia1 d-polytopes with u vertices and a ( k - l)-face F such that f O ( l k I c P=~ )r - k, we have 6)

min hi (Ikxcp,F)=

(ii)

min hi ( P ) =

r

u-d,

(iii)

minh,(X(P)\F) =

i =0, rl'- d ,

lSiSm; i =0, l ~ i s n ;

1, v - d, v-d -1, v - r, 0,

i =0, 1 s i s k - 1, i = k, k + 1s i G d -1, i =d;

Bounding the numbers of faces of polytope pairs

(iv)

minfi(Ikr(p,F)=f,(P(r - k,d - k ) ) ,

(v)

minfi (P) = fi ( P ( v ,d ) ) , 0 s j

(vi)

m i n i ( z ( P ) \ F ) = fi ( P ( v ,d)) - f i - k ( P ( r - k, d - k ) ) ,

(vii)

max h, (Ikz@)

G

(viii) max hi (P)s ( v - d + i - 1) - ( v - d

(ix)

O S j s d - k -1;

d - 1;

,

=

229

0 sj s d

1;

OSiSm;

+ i - k -1

i-k

max hi ( z ( P ) \ F ) s hi ( C ( Vd, ) ) - hi-k ( C ( V-2k, d - 2k))

+ hi-k ( P ( r -2k, d - 2 k ) ) - hi-k (P(r - k, d - k ) ) , (x)

max fi (lkr(p,F)= fi (C(r - k, d - k )),

(xi)

maxfi(P)Gfi(C(v,d)) i =O

0S j

S

Os i
d - k - 1;

(f) [f,-k-i+l(C(vA2k,d-2k))

-fi-k-i+l(c(r - 2k, d - 2k))], (xii)

-

0 S j s d - 1;

maxfi(z((P)\F)cfi(C(v,d))-fi-l,(P(r - k,d - k ) )

-i(F) 1 =O

[fi-k-i+l(c(v - 2 k , d -2k))

--fi-k-i+I(P(r - 2k, d - 2k))],

+

0sj s d

- 1,

with equality in (viii) and (ix) if 0 s i 6 [ ( d - k 1)/2] or [ ( d + k)/2] s i s d and equality in (xi) and (xii) if 0 S j G [(d - k + 1)/2] - 1. Moreover, equality holds in (viii), (ix), (xi) and (xii) if k [ ( d - 1)/2], and equality holds in (viii) and (xi) if r = v. Remark. As with Theorem 4,some of the inequalities above simplify if k 3 n :

m a x h i ( s ( P ) \ F ) s hi(C(v,d))-hi-k(P(rmax fi (P)s h ( C (v, d)),

0
k,d - k ) ) ,

0 si sd ;

6 d - 1;

and max fi (T,(P)\F ) s fi (C(v, d)) - h - k ( P ( r - k, d - k)),

0sj sd

- 1.

C.W. Lee

230

Proof. Because fo(lk,,,)F) = r - k we know that hl(lkr(,,F) = r - d. Define h ! ” = h , ( P ( r - k , d - k ) ) , g ! ” = h ! ” - h ! ” l and h ! * ’ = h , ( C ( r - k , d - k ) ) f o r a l l i. Let h E h ( P t ) with h , = r - d, and put h, = 0 if i < 0 or i > d. By Theorem 1, h, 3 h?),0 G i C m, which gives us (i) and (ii) using Theorem 4. For 1S i S m. h, -ht-k = ( h , - h , - ~ ) + ( h , - ~ - h , - z ) + . . . + ( h , - k + l - h ~ - k )

g!”+ g!!I + * - h!l)- h(1) I-kr

3

*.

+ g!?k+l

and for m + 1 S i S d -rn - 1 , the unimodality of h forces h, - h,-k2 h t ) - h!!!k.Thus we obtain (iii) using Theorem 4. By Theorem 1 or the Upper Bound Theorem, h, s hy’, O G i c m, giving us (vii) and (viii) using Theorem 4. For (ix) we use Theorem 4 (iv) and attempt to make hd-s-k - h,-k as large as possible, i.e. h,-k - hd-,-k as small as possible, n + l G i S d . If n + l < i i d - m - l , then i - k G d - k - m - l s m and i - k > d - i - k, so

= h?) - h(1) I-k

d-i-k.

If, on the other hand, d - m S i S d, then d - i - k < d - i

2

d-i

=

j=d-i-k+l

c

d-i

2

G

m, so

(h, - hj-1) gy

j=d-i-k+l

Therefore we obtain (ix). The minima and the maxima for the f;. in (iv), (v), (vi) and (x) are determined from the facts that the f, are non-negative linear combinations of the hi,and that in each of these cases the bounds on the hi are simultaneously achievable. The upper bounds in (xi) and (xii) are similarly obtained from the upper bounds in (viii) and (ix). Equality in (viii) and (ix) if 0 s i s [ ( d - k + 1)/2] or [ ( d + k)/2] =si =sd

Bounding the numbers of faces of polytope pairs

23 1

comes from case (1) of Theorem 4, from which the equalities in (xi) and (xii) for 0 s j S [(d - k + 1)/2] - 1 follow, recalling that f i is a non-negative linear combination of ho, hl, . . . ,h,,,. Equality for (viii) and (xi) if r = v follows by case (2) of Theorem 4 and equality for (viii), (ix), (xi) and (xii) if k S [(d - 1)/2] follows from case (3) of Corollary 2. Corollary 4. Let 4 6 d < r s v and 2 s k s d - 2. As P ranges over all simple d-polyhedra with recession cone of dimension d - k + 1, and v facets, r of which are unbounded, then (i)

minfi ( p ) = fd-j-l(P(v, d)) - fd-j-k-i(P(T - k, d - k)),

(ii)

m a X f ; ( P ) ~ f d - j - i ( C ( V , d ) ) - f d - j - - k - - I ( P ( rk,d -

-

-i(F) [fd-i-j-k(C(V-2k,d-2k)) I

0sj

d - 1;

k))

=O

-fd-i-j-k(P(r-2k,d

-2k))],

0SjSd-1.

Moreover, there exists a simple d-polyhedron PI satisfying the above conditions such that f(P,)achieves all of the values in (i); and if k S [(d - 1)/2], there exists a simple d-polyhedron P2satisfying the above conditions such that f(P,) achieves all of the upper bounds in (ii). Proof. Immediate from Corollary 3 (vi) and (xii) and duality. Whether the inequalities of Theorem 4 (iii) and (iv); Corollary 3 (viii), (ix), (xi) and (xii); and Corollary 4 (ii) are actually equalities remains to be seen. 6. h-Vectors of polyhedral balls

McMullen’s conditions provide a characterization of the set of h-vectors of simplicia1 polytopes, but as yet there is no characterization of the h-vectors of polyhedral balls. Two necessary conditions for a vector of integers h = (ho,hl, . . . ,hd) to be the h-vector of Z ( P ) \ F for some simplicia1 d-polytope P with (k - 1)-face F are: (1) (ho- hd+j,hl - hd+j-l,.. . ,h, - hd+j-r) is an 0-sequence for all integer 0 s j s d + 1, where r = [(d + j - 1)/2]. (2) There exist vectors a = (ao,a l , .. . ,a d - k ) E h ( P f - 3 and b = (bo,bl,. .. ,b d ) f h(9:’) such that hi + ai-n = bi,0 s i S d. Condition (2) is immediate from the discussion in the beginning of Section 5 and the proof of condition (1) is the same as that in [5],once it is observed that the boundary of a polyhedral (d - 1)-ball is always a polyhedral (d - 2)-sphere.

232

C.W. Lee

Conjecture 1. Let h = (ho,h,, . . .,h,) be a (d + 1)-vector of integers. Then h = h ( X ( P ) \ F ) for some simplicial d-polytope P with (k - 1)-face F, 1S k S d, if and only if h satisfies (1) and (2). We remark that the conjecture is true for k = d - 1 or k = d using the sufficiency of McMullen’s conditions for h -vectors of simplicial d -polytopes, and that condition (2) is implied by condition (1) in the case that k = 1, using [5, Corollary 3.91.

Note added in proof. A construction of Barnette has established that all upper bounds in Corollaries 3 and 4 are tight.

References [l] D.W. Barnette, The minimum number of vertices of a simple polytope, Israel J. Math. 10 (1971)

121-125. [2] D.W. Barnette, A proof of the lower bound conjecture for convex polytopes, Pacific J. Math. 46 (1973) 349-354. [3] L.J. Billera and C.W. Lee, Sdiciency of McMullen’s conditions for f-vectors of simplicial polytopes, Bull. (New Series) Amer. Math. SOC.2 (1980) 181-185. 14) L.J. Billera and C.W. Lee, A proof of the suf6ciency of McMullen’s conditions for f-vectors of simplicial convex polytopes, J. Combinat. Theory (A) 31 (1981) 237-255. (51 L.J. Billera and C.W. Lee, The numbers of faces of polytope pairs and unbounded polyhedra, European J. Combinatorics 2 (1981) 307-322. [6] B. Grunbaum, Convex Polytopes (Wiley, New York, 1967). (71 V. Klee, A comparison of primal and dual methods for linear programming, Numer. Math. 9 (1966) 227-235. [8l V.n e e , Polytope pairs and their relationship to linear programming, Acta Math. 133 (1974) 1-25. [9] C.W.Lee, Counting the faces of simplicia1 convex polytopes, Ph.D. Thesis, Cornell University, Ithaca, N.Y. (1981). [lo] P. McMullen, The maximum numbers of faces of a convex polytope, Mathematika 17 (1970) 179-184. [ll] P. McMullen and G.C. Shephard, Convex polytopes and the upper bound conjecture, London Math. SOC.Lecture Note Series 3 (Cambridge, 1971). 1121 J.S. Provan and L.J. Billera, Decompositions of simplicial complexes related to diameters of convex polyhedra, Math. Oper. Res. 5 (1980) 576-594. [13] R.P. Stanley, The number of faces of a simplicial convex polytope, Adv. Math. 35 (1980) 236-238.

Received 13 April 1981 ; revised 25 October 1981