andKonstantyJunosza-Szaniawski AdamIdzik COMBINATORIALLEMMASFORPOLYHEDRONSI GraphTheory26 ( 2006 ) 439–448 DiscussionesMathematicae 439

COMBINATORIAL LEMMAS FOR POLYHEDRONS I

Adam Idzik Akademia ´ Swi¸etokrzyska

Swi¸etokrzyska 15, 25–406 Kielce, Poland ´ and

Institute of Computer Science, Polish Academy of Sciences Ordona 21, 01–237 Warsaw, Poland

e-mail: adidzik@ipipan.waw.pl and

Konstanty Junosza-Szaniawski Warsaw University of Technology Pl. Politechniki 1, 00–661 Warsaw, Poland

e-mail: k.szaniawski@mini.pw.edu.pl

Abstract

We formulate general boundary conditions for a labelling of vertices of a triangulation of a polyhedron by vectors to assure the existence of a balanced simplex. The condition is not for each vertex separately, but for a set of vertices of each boundary simplex. This allows us to formulate a theorem, which is more general than the Sperner lemma and theorems of Shapley; Idzik and Junosza-Szaniawski; van der Laan, Talman and Yang. A generalization of the Poincar´e-Miranda theorem is also derived.

Keywords: b-balanced simplex, labelling, polyhedron, simplicial complex, Sperner lemma.

2000 Mathematics Subject Classification: 05B30, 47H10, 52A20,

54H25.

1. Preliminaries

For n ∈ N, let N = {1, . . . , n} and N ₀ = {0, . . . , n}. By a polyhedron we understand the convex hull of a finite set of R ⁿ . Let P ⊂ R ⁿ be a polyhedron of dimension n. A face of the polyhedron P is the intersection of P with some of its supporting hyperplanes. Denote the set of all k- dimensional faces of the polyhedron P by F _k (P ) (k < n), the set of all faces of the polyhedron P by F(P ) (hence F(P ) = S _n−1

k=0 F _k (P )) and the set of all vertices of the polyhedron P by V (P ) (V (P ) = F ₀ (P )). The maximal dimension proper faces of the polyhedron P are called facets. For a finite set A = {a ₀ , . . . , a _m } ⊂ R ⁿ a set co A = {α ₀ a ₀ + · · · + α _m a _m : a _i ∈ A, P _m

i=0 α _i = 1, α _i ≥ 0 for i ∈ {0, . . . , m}} is the convex hull of A, aff A = {α ₀ a ₀ + · · · + α _m a _m : P _m

i=0 α _i = 1, a _i ∈ A, α _i ∈ R for i ∈ {0, . . . , m}}

is the affine hull of A. And if for a finite set A = {a ₀ , . . . , a _m } ⊂ R ⁿ (m ∈ {0, . . . , n}) the dimension of aff A is equal to m, then co A is called a simplex (precisely an m-simplex). Let T r _n be a finite family of n-simplexes such that P = S

δ∈T r

δ and for any δ ₁ , δ ₂ ∈ T r _n , δ ₁ ∩ δ ₂ is the empty set or their common face. A triangulation of the polyhedron P (we denote it by T r) is a family consisting of simplexes of T r _n and all their faces. Let T r _m (m ∈ N ₀ ) denote the family of m-simplexes belonging to a triangulation T r.

Hence T r = S _n

i=0 T r _i . Let V = T r ₀ be the set of vertices of all simplexes of T r. Notice, that V = S

δ∈T r

V (δ). An (n − 1)-simplex of T r _n−1 is a boundary (n − 1)-simplex if it is a facet of exactly one n-simplex of T r _n . For a triangulation T r ^P of the polyhedron P and a triangulation T r ^Q of a polyhedron Q a function f : V (T r ^P ) → V (T r ^Q ) is a simplicial function if for every σ ∈ T r ^P there exists δ ∈ T r ^Q such that f (V (σ)) = V (δ).

2. Main Result We start with the following

Definition 2.1. Let σ ⊂ R ⁿ be a simplex, l : V (σ) → R ⁿ , b ∈ R ⁿ and Z ⊂ R ⁿ . A simplex σ is b-balanced if the point b belongs to co (l(V (σ))) and b-subbalanced with respect to Z, if the point b belongs to co (l(V (σ)) ∪ Z).

If Z = {x}, then we write b-subbalanced with respect to x instead of with

respect to {x}. For b = 0 we say balanced and subbalanced instead of

b-balanced and b-subbalanced, respectively.

Notice that in the case Z is a polyhedron, a simplex σ is b-subbalanced with respect to Z if and only if σ is b-subbalanced with respect to V (Z).

Lemma 2.2. Let P ⊂ R ⁿ be a polyhedron of dimension n, T r be a tri- angulation of the polyhedron P , l : T r ₀ → R ⁿ , b ∈ R ⁿ and x ∈ R ⁿ . If the triangulation T r contains neither a b-balanced simplex of dimension less than n nor a simplex of dimension less than n − 1 which is b-subbalanced with respect to x, then the number of b-balanced simplexes in T r is con- gruent modulo 2 to the number of b-subbalanced with respect to x boundary simplexes in T r.

P roof. For this proof by a b-subbalanced simplex we understand a b- subbalanced simplex with respect to x. Consider a graph G = (W, E) where W is the set of b-balanced n-simplexes and b-subbalanced (n − 1)-simplexes in T r and there is an edge between two different simplexes σ ₁ , σ ₂ ∈ W if and only if there exists a simplex σ ∈ T r containing σ ₁ and σ ₂ (in particular σ = σ ₁ ). We will show that

deg _G (σ) =

( 1 if σ is a b-balanced or a boundary b-subbalanced simplex, 2 if σ is a b-subbalanced simplex not in the boundary.

Let σ be a b-balanced simplex of T r. By our assumption σ is an n-dimen- sional simplex. Let V (σ) = {v ₀ , . . . , v _n }, u _i = l(v _i ) for i ∈ N ₀ and let A _i = co {u ₀ , . . . , u _i−1 , x, u _i+1 , . . . , u _n } for i ∈ N ₀ . There is at least one j ∈ N ₀ such that b ∈ A _j since b ∈ co {u ₀ , . . . , u _n } ⊆ S _n

i=0 A _i . If there exists j, k ∈ N ₀ , j < k such that b ∈ A _j and b ∈ A _k , then it is easy to show that b ∈ co {x, u ₀ , . . . , u _j−1 , u _j+1 , . . . , u _k−1 , u _k+1 , . . . , u _n }, so that the simplex co {v ₀ , . . . , v _j−1 , v _j+1 , . . . , v _k−1 , v _k+1 , . . . , v _n } is b-subbalanced and of dimension less than n − 1. This contradicts our assumption.

Now let σ be a b-subbalanced simplex in T r of dimension n − 1 and let σ ₁ be an n-simplex containing σ, V (σ) = {v ₁ , . . . , v _n }, V (σ ₁ ) \ V (σ) = {v ₀ }, u _i = l(v _i ) for i ∈ N ₀ , B ₀ = co {u ₀ , u ₁ , . . . , u _n }, B _i = co {x, u ₁ , . . . , u _i−1 , u ₀ , u _i+1 , . . . , u _n } for i ∈ N ₀ . Since b ∈ co {x, u ₁ , . . . , u _n } ⊆ S _n

i=0 B _i , then

there exists i ∈ N ₀ such that b ∈ B _i . If b ∈ B ₀ , then σ ₁ is b-balanced

and σ and σ ₁ form an edge in G. If b ∈ B _i for some i ∈ N , then σ ₂ =

co {v ₀ , . . . , v _i−1 , v _i+1 , . . . , v _n } is b-subbalanced and σ and σ ₂ form an edge in

G. If b ∈ B ₀ ∩ B _j for some j ∈ N , then the simplex co {v ₁ , . . . , v _j−1 , v _j+1 ,

. . . , v _n } is b-subbalanced of dimension less that n − 1, but this is im-

possible. If b ∈ B _j ∩ B _k for some j, k ∈ N , j < k, then the simplex

co {v ₀ , . . . , v _j−1 , v _j+1 , . . . , v _k−1 , v _k+1 , . . . , v _n } is b-subbalanced of dimension less that n − 1, but this is also impossible. In all cases, σ ₁ defines an adja- cent edge to σ in G. Hence, if σ is a boundary simplex (it is a face of exactly one n-simplex), then deg _G (σ) = 1 and if σ is not a boundary simplex (it is a face of exactly two n-simplexes), then deg _G (σ) = 2.

Graph G has vertices of degree one and two only. Thus the number of vertices of degree one is even and hence the number of b-balanced simplexes in T r is congruent modulo two to the number of b-subbalanced with respect to x boundary simplexes in T r.

Remark 2.3. Let S ⊂ R ⁿ be a polyhedron, f T r be a triangulation of bd S and p ∈ ri S, then T r = {co({p} ∪ σ) : σ ∈ f T r} ∪ f T r ∪ {p} is a triangulation of the polyhedron S.

Definition 2.4. Two n-dimensional polyhedrons P and Q are dual to each other through ψ if ψ : F(P ) → F(Q) is a one-to-one inclusion-reversing mapping, i.e., F ₁ ⊂ F ₂ if and only if ψ(F ₁ ) ⊃ ψ(F ₂ ) for any F ₁ , F ₂ ∈ F(P ).

Polyhedrons P and Q are dual to each other if there exists ψ : F(P ) → F(Q) such that P and Q are dual to each other through ψ.

A simplex of any dimension is dual to itself and a 3-dimensional cube and octahedron are dual to each other. For more examples and properties of dual polyhedrons see Grunbaum [4], pp. 46–48. Notice that dim F + dim ψ(F ) = n − 1 for any F ∈ F(P ).

Duality of polyhedrons may be defined in many ways (see e.g. Alexan- drov [1], pp. 49):

Definition 2.5. Two n-dimensional polyhedrons P and Q are dual to each other through φ, if φ : F ₀ (P ) → F _n−1 (Q) fulfils the following condition: for v ₁ , v ₂ ∈ F ₀ (P ), co {v ₁ , v ₂ } is a face of P if and only if φ(v ₁ ) and φ(v ₂ ) have a common (n − 2)-dimensional face.

Observe that both definitions are equivalent.

Theorem 2.6. Let P, Q ⊂ R ⁿ be n-dimensional polyhedrons, dual to each

other through a mapping ψ, T r be a triangulation of the polyhedron P , V =

T r ₀ , b ∈ ri Q and l : V → R ⁿ be a labelling. If for every G ∈ F(P ) and

every simplex σ ∈ T r and σ ⊆ G, σ is not b-subbalanced with respect to the

set ψ(G), then there exists a b-balanced simplex in T r.

P roof. For n = 1 the boundary condition implies that the labels of two vertices of P lie on opposite sides of the point b. Thus there is a vertex v ∈ T r ₀ such that l(v) = b or the number of b-balanced simplexes in T r is odd.

Consider the case n > 1. Assume that there is no b-balanced simplex in T r of dimension less than n. We show that there exists a b-balanced simplex of dimension n in T r.

We define a triangulation T r ^Q of bd Q. For every face H ∈ F ₁ (Q) we choose a point u _H ∈ ri H and apply Remark 2.3 to get a triangulation of the face H. Then inductively for k = 2, . . . , n − 1: for every face H ∈ F _k (Q) we choose a point u _H ∈ ri H and apply Remark 2.3 to get a triangulation of the face H. Finally we obtain a triangulation of bd Q.

Let V (P ) = {a ₀ , . . . , a _k } (k ≥ n). For i ∈ {0, . . . , k} and c ∈ ri P , let a ⁰ _i = 2a _i − c and P ⁰ = co {a ⁰ ₀ , . . . , a ⁰ _k }. Notice that P ⊂ P ⁰ .

Now we define a triangulation of P ⁰ , which is an extension of the tri- angulation T r of the polyhedron P . We define a triangulation of the set P ⁰ \ ri P .

For every face F = co {a _i(0) , . . . , a _i(l) } (defined by some: {a _i(0) , . . . , a _i(l) }

⊂ V (P )) of the polyhedron P we denote F ⁰ = co {a ⁰ _i(0) , . . . , a ⁰ _i(l) }. Every face F of P has one-to-one correspondence to the face F ⁰ of P ⁰ .

Let us denote F F ⁰ = co {F ∪ F ⁰ }. Thus P ⁰ \ ri P = S

F ∈F

(P ) F F ⁰ . For every face F ₁ ∈ F ₁ (P ) we choose a point v _F

∈ ri F ₁ ⁰ . By Remark 2.3 we receive a triangulation of F ₁ ⁰ . Then for every face F ₁ ∈ F ₁ (P ) we choose a point v _F

_F

1

∈ ri F ₁ F ₁ ⁰ . By Remark 2.3 we receive a triangulation of F ₁ F ₁ ⁰ .

Now we apply the induction for k ∈ {2, . . . , n − 1}: for any face F _k ∈ F _k (P ) we choose a point v _F

k

∈ ri F _k ⁰ and by Remark 2.3 we get a triangula- tion of the face F _k ⁰ . Analogously we choose a point v _F

_F

k

∈ ri F _k F _k ⁰ and get a triangulation of F _k F _k ⁰ .

Finally we obtain a triangulation of P ⁰ \ ri P and denote it by T r ⁰⁰ . Hence T r ⁰ = T r ∪ T r ⁰⁰ is a triangulation of P ⁰ , which is an extension of the triangulation T r on P .

Let V ⁰ = T r ₀ ⁰ . If v ∈ V ⁰ \ V , then v ∈ V (P ⁰ ) ∪ {v _GG

, v _G

: G ∈ F _k (P ), k ∈ {1, . . . , n − 1}}. For G ⁰ ∈ F ₀ (P ⁰ ) we also denote v _G

:= G ⁰ .

Now, we define a labelling l ⁰ : V ⁰ → R ⁿ :

l ⁰ (v) = (

l(v) for v ∈ V,

u _ψ(G) for v = v _GG

or v = v _G

.

We prove that there is no b-balanced simplex in T r ⁰⁰ . Consider an n-simplex σ ∈ T r ⁰⁰ . If σ ∩ P = ∅, then there is exactly one vertex v of σ, which is also a vertex of P ⁰ . Let v = a ⁰ _j ∈ V (P ⁰ ) (j ∈ N) and thus l ⁰ (V (σ)) ⊂ ψ(a _j ), where ψ(a _j ) is a facet of Q so σ is not b-balanced. Now consider the case σ ∩P 6= ∅:

let τ = σ ∩ P , G _τ be the smallest face of P (in the sense of inclusion) containing τ . Let v ∈ V (σ) \ V (τ ), thus v = v _G

or v = v _GG

for some G ∈ F(P ). Notice that G _τ ⊆ G and thus ψ(G _τ ) ⊇ ψ(G). From definition of T r ^Q we have u _H ∈ H for any H ∈ F(Q) and from definition of labelling l ⁰ we have l ⁰ (v) = u _ψ(G) ∈ ψ(G) ⊆ ψ(G _τ ). Thus l ⁰ (V (σ) \ V (τ )) ⊆ ψ(G _τ ) and l ⁰ (V (σ)) = l ⁰ ((V (σ) \ V (τ )) ∪ V (τ )) = l ⁰ ((V (σ) \ V (τ ))) ∪ l ⁰ (V (τ )) ⊆ ψ(G _τ ) ∪ l ⁰ (V (τ )) = ψ(G _τ ) ∪ l(V (τ )).

From the assumption b / ∈ co (l(V (τ ))∪ψ(G _τ )). Therefore b / ∈ co l ⁰ (V (σ)) and σ is not a b-balanced simplex.

Let σ be an (n − 1)-simplex, V (σ) = {v ₁ , . . . , v _n }, v ⁰ _i = 2b − v _i (i ∈ N ), C(σ) = cone({v ₁ ⁰ , . . . , v ⁰ _n }, b). An (n − 1)-simplex σ is b-subbalanced with respect to x if and only if x ∈ C(σ). The set C(σ) is an (n − 1)-dimensional set and the union S

σ∈T r

,σ⊂bd P

C(σ) is also an (n − 1)-dimensional set.

Hence, we can choose x ∈ R ⁿ , x 6= b in such a way that T r ⁰ does not contain a b-subbalanced simplex with respect to x of dimension smaller that n − 1. Consider a line going through x and b. This line meets bd Q in two points. By x ⁰ we denote the common point of this line and bd Q such that b ∈ co {x, x ⁰ } and by σ _Q ∈ T r ^Q we denote the (n − 1)-dimensional boundary simplex containing x ⁰ . The function l ⁰ restricted to the set bd P ⁰ ∩ V ⁰ is a one-to-one simplicial function. The simplex σ _P := co l ⁰⁻¹ (V (σ _Q )) is b- subbalanced with respect to x and it is the only such simplex on bd P .

Now, from Lemma 2.2 it follows that the number of b-balanced simplexes in T r ⁰ is odd. Since T r ⁰ = T r ∪ T r ⁰⁰ and there is no b-balanced simplex in T r ⁰⁰ , there exists b-balanced simplex in T r.

3. Corollaries and Applications

In this section we present corollaries to Theorem 2.6 in order to show the strength of this theorem. First we apply Theorem 2.6 to the simplex:

Corollary 3.1. Let P = co {d ₀ , . . . , d _n } ⊂ R ⁿ be an n-dimensional simplex, m _F

= P

j6=i d

n be the gravity center of a facet F _i = co {d ₀ , . . . , d _i−1 , d _i+1 , . . . , d _n }, m _P = P _n

j=0 d

n+1 be the gravity center of P , T r be a triangulation

of the simplex P , V = T r ₀ and l : V → R ⁿ be a labelling. If for every face F = co {d _i : i ∈ M } and every simplex σ ⊂ F , σ ∈ T r, σ is not m _P -subbalanced with respect to the set {m _F

: i ∈ M }, then there exists an m _P -balanced simplex in T r.

Corollary 3.1 is more general than the Sperner lemma [12] and the Shapley lemma (Lemma 7.2 in [11]).

Corollary 3.2. Let P = co {d ₀ , . . . , d _n } ⊂ R ⁿ be an n-dimensional simplex, m _P = P _n

j=0 d

n+1 be the gravity center of P , T r be a triangulation of the simplex P , V = T r ₀ and l : V → R ⁿ be a labelling. If for every face F = co {d _i : i ∈ M } (M ⊂ N ₀ ) and every simplex σ ⊂ F , σ ∈ T r, σ is not m _P -subbalanced with respect to the set {d _i : i / ∈ M }, then there exists an m _P -balanced simplex in T r.

Corollary 3.2 is more general than the Scarf lemma ([10]; see also Theorem 3.4 in [9]) and the Garcia lemma ([3], see also Theorem 3.6 in [9]).

The next result is on an n-dimensional cube. Let I ⁿ = {(x ₁ , . . . , x _n ) ∈ R ⁿ : −1 ≤ x _i ≤ 1, i ∈ N } be an n-dimensional cube and for k ∈ N , i ₁ , . . . , i _k ∈ N , i ₁ < i ₂ < · · · < i _k , s _i

, . . . , s _i

∈ {−1, 1} let I(s _i

i ₁ , . . . , s _i

i _k )

= {(x ₁ , . . . , x _n ) ∈ I ⁿ : x _i

= s _i

, j ∈ {1, . . . , k}} be an (n − k)-dimensional face of I ⁿ .

Corollary 3.3. Let T r be a triangulation of the cube I ⁿ , V = T r ₀ and l : V → R ⁿ be a labelling. If for all k ∈ N , i ₁ , . . . , i _k ∈ N , i ₁ < i ₂ < · · · < i _k , s _i

, . . . , s _i

∈ {−1, 1} and every simplex σ ∈ T r and σ ⊆ I(s _i

i ₁ , . . . , s _i

i _k ), σ is not subbalanced with respect to the set {s _i

e _i

: j ∈ {1, . . . k}}, then there exists a balanced simplex in T r.

P roof. It follows directly from Theorem 2.6 for P = I ⁿ , Q = co{e _i , −e _i : i ∈ N } and ψ({s _i

e _i

: j ∈ {1, . . . , k}}) = co {s _i

e _i

: j ∈ {1, . . . , k}} for all k ∈ N , i ₁ , . . . , i _k ∈ N , i ₁ < i ₂ < · · · < i _k , s _i

, . . . , s _i

∈ {−1, 1}.

Corollary 3.3 is more general than the Freund lemma (Lemma 1 in [2], see also Lemma 3.7 in [9]). Our next result is a generalization of the Poincar´e- Miranda theorem [8]:

Theorem 3.4. Let f : I ⁿ → R ⁿ be a continuous function, such that for all

k ∈ N , i ₁ , . . . , i _k ∈ N , i ₁ < i ₂ < · · · < i _k , s _i

, . . . , s _i

∈ {−1, 1}

f (I(s _i

i ₁ , . . . , s _i

i _k )) ∩ cone{−s _i

e _i

: j ∈ {1, . . . , k}} = ∅, then there exists x ∈ I ⁿ such that f (x) = 0.

P roof. Consider a sequence of triangulations T r ^m of I ⁿ (m ∈ N) with mesh tending to zero, when m tends to infinity. Let V ^m = V (T r ^m ) and l ^m = f | _V

. We show that the labelling l ^m fulfils the condition of Corollary 3.3. Let I(s _i

i ₁ , . . . , s _i

i _k ) be a face of I ⁿ for k ∈ N , i ₁ , . . . , i _k ∈ N , i ₁ < i ₂ <

· · · < i _k , s _i

, . . . , s _i

∈ {−1, 1}. Take σ ^m ⊆ I(s _i

i ₁ , . . . , s _i

i _k ). Because f is continuous and for sufficiently large m the mesh of T r ^m is small enough, the condition f (I(s _i

i ₁ , . . . , s _i

i _k )) ∩ cone{−s _i

e _i

: j ∈ {1, . . . , k}} = ∅ implies l ^m (V (σ ^m )) ∩ cone{−s _i

e _i

: j ∈ {1, . . . , k}} = ∅. This is equivalent to the condition that σ ^m is not subbalanced with respect to the set {s _i

e _i

: j ∈ {1, . . . , k}}. Corollary 3.3 implies that there exists a balanced simplex in T r ^m . Now, if the mesh of T r ^m tends to zero, the sequence of simplexes σ ^m tends to a point z. But each σ ^m is a balanced simplex so we have f (z) = 0.

Theorem 2.6 is more general than our previous result:

Corollary 3.5 (Theorem 3.4 in [6]). Let P ⊂ R ⁿ be a polyhedron of dimen- sion n, T r be a triangulation of the polyhedron P , V = T r ₀ , b ∈ ri P and l : V → R ⁿ be a labelling. If for every facet F of the polyhedron P there ex- ists an (n−1)-dimensional hyperplane h ^F _b containing the point b and disjoint with F such that l(V ∩ F ) ⊂ H _b ^F , where H _b ^F is an open halfspace containing F such that h ^F _b is in its boundary, then there exists a b-balanced n-simplex in the triangulation T r.

P roof. For every polyhedron P and any point b ∈ P there exists a dual polyhedron Q such that every face of Q is perpendicular to the ray issu- ing from b through the vertices of P and whose vertices lie on the rays issuing from b and perpendicular to the faces of P (for the proof see [1], pp. 45). Hence P, Q and l fulfil the condition of Theorem 2.6 and we get our corollary.

For every n-dimensional polyhedron P ⊂ R ⁿ and any point x ₀ ∈ ri P there exist vectors a _i (i ∈ I ⊂ N), such that P = {x ∈ R ⁿ : a _i x ≤ 1 + a _i x ₀ , i ∈ I}.

Let car B = {i ∈ I : a _i x = 1 + a _i x ₀ for all x ∈ B} for B ⊂ P .

Two theorems below, proved by van der Laan, Talman and Yang, follow

also from our Theorem 2.6:

Corollary 3.6 (Theorem 4.1 in [9]). Let P = {x ∈ R ⁿ : a _i x ≤ 1 + a _i x ₀ , i ∈ I} ⊂ R ⁿ be an n-dimensional polyhedron, Q = co {a _j ∈ R ⁿ : i ∈ I}, b ∈ ri Q, T r be a triangulation of P , V = T r ₀ and l : V → R ⁿ be a labelling.

There exists a simplex σ ∈ T r such that b ∈ co (l(V (σ)) ∪ {a _i : i ∈ car σ}).

P roof. For a polyhedron P = {x ∈ R ⁿ : a _i x ≤ 1 + a _i x ₀ , i ∈ I} ⊂ R ⁿ the polyhedron Q = co {a _j ∈ R ⁿ : i ∈ I} is dual to P through a mapping ψ : F(P ) → F(Q) defined by ψ(F ) = co {a _i ∈ R ⁿ : i ∈ car F }, (for the proof see Grunbaum [4] pp. 46–49).

Notice that for any boundary simplex σ ∈ T r the condition b ∈ co (l(V (σ)) ∪ {a _i : i ∈ car σ}) says that σ is b-subbalanced with respect co {a _i ∈ R ⁿ : i ∈ car σ}. Hence, if there is no boundary simplex σ such that b ∈ co (l(V (σ)) ∪ {a _i : i ∈ car σ}), then the assumptions of Theorem 2.6 are satisfied and we get the thesis.

For any k-dimensional polyhedron P ⊂ R ⁿ there exists m vectors a _i ∈ R ⁿ , m real numbers α _i ∈ R (m > k, i ∈ I ⊂ N), and n − k vectors d _h ∈ R ⁿ , n − k real numbers δ _h ∈ R (h ∈ N _k ⊂ N) such that P = {x ∈ R ⁿ : a _i x ≤ α _i for i ∈ I, d _h x = δ _h for h ∈ N _k }.

Corollary 3.7 (Theorem 3.1 in [9]). Let P = {x ∈ R ⁿ : a _i x ≤ α _i for i ∈ I, d _h x = δ _h for h ∈ N _k } be a k-dimensional polyhedron, W = aff {d _h : h ∈ N _k }, W ^∗ = {x ∈ R ⁿ : xy = 0 for all y ∈ W }, T r a triangulation of P , l : V → R ⁿ be a labelling such that (co l(V )) ∩ W = {0}. If for every F ∈ F(P ) and every simplex σ ∈ T r, σ ⊆ F the intersection (co l(V (σ))) ∩ (cone(0, {a _i : i ∈ car (F )}) + W ) is empty or contains the point 0 ∈ R ⁿ , then there exists a balanced simplex in T r.

P roof. Without loss of generality we may assume that the vectors a _i for

i ∈ I are parallel to the hyperplane W ^∗ . We can consider projection of

the polyhedron P , labels l(V ) and vectors a _i for i ∈ I on the hyperplane

W ^∗ parallel to the hyperplane W . Hence, we reduce this theorem to the

full-dimensional case. Analogously vectors a _i (i ∈ I) can be scaled in such a

way that Q = co {a _i : i ∈ I} is a polyhedron dual to P through a mapping

ψ : F(P ) → F(Q) defined by ψ(F ) = co {a _i ∈ R ⁿ : i ∈ car F }. If there exists

a simplex σ such that co l(V (σ)) contains 0 ∈ R ⁿ , then σ is a balanced sim-

plex. If for any F ∈ F(P ) and any simplex σ ∈ T r, σ ⊆ F the intersection

co l(V (σ)) ∩ cone(0, {a _i : i ∈ car (F )}) is empty, then σ is not a subbal-

anced simplex with respect to the set ψ(F ) and by Theorem 2.6 we get our

theorem.

Acknowledgment

The authors thank the anonymous referee for many helpful comments and suggestions.

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Received 2 December 2005

Revised 3 October 2006