A NEW TOPOLOGICAL HELLY THEOREM AND

A NEW TOPOLOGICAL HELLY THEOREM AND SOME TRANSVERSAL RESULTS L. MONTEJANO Abstract. We prove that for a topological space X with the property that H (U ) = 0 for d and every open subset U of X, a …nite family of open sets in X has nonempty intersection if for any subfamily of size j; 1 j d+1; the (d j)-dimensional homology group of its intersection is zero. We use this theorem to prove new results concerning transversal a¢ ne planes to families of convex sets

1. Introduction and Preliminaries A prominent role in combinatorial geometry is played by the Helly Theorem [7], which states that a …nite family of convex sets in Rd has nonempty intersection if and only if any subfamily of size at most d+1 has nonempty intersection. Helly himself realized in 1930 (see [9]) that a …nite family of sets in Rd has nonempty intersection if for any subfamily of size at most d + 1; its intersection is homeomorphic to a ball in Rd . In fact, the result is true if we replace the notion of topological ball by the notion of acyclic set, see [3] and [10]. In 1970, Debrunner [8] proved that a …nite family of open sets in Rd has nonempty intersection if for any subfamily of size j; 1 j d + 1; its intersection is (d j)-acyclic. Our theorem follows the same spirit except that instead of Rd , we require a topological space X in which H (U ) = 0 for d and every open subset U of X: Moreover, instead of the hypothesis (d j)-acyclic, we just require that the (d j)-dimensional homology group is zero. That is, we prove that for a topological space X with the property that H (U ) = 0; for d and every open subset U of X; a …nite family of open sets in X has nonempty intersection if for any subfamily of size j; 1 j d + 1; the (d j)-dimensional homology group of its intersection is zero, where H 1 (U ) = 0 if and only if U is nonempty. The fact that this is a non-expensive topological Helly theorem — in the sense that it does not require the open sets to be simple— from the homotopy point of view (we only require its (d 1)-dimensional 2000 Mathematics Subject Classi…cation. Primary 52A35,55N10. Key words and phrases. Helly theorem, homology group, transversal. This research is supported by CONACYT, 41340. 1

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homology group to be zero), allows us to prove interesting new results concerning transversal planes to families of convex sets. During this paper, we use reduced singular homology with any nonzero coe¢ cient group. Let U be a topological space. We say that H 1 (U ) = 0 if and only if U is nonempty and we say that U is connected if and only if H0 (U ) = 0. At the levels q = 1 and q = 0, we mean that the Mayer-Vietoris sequence is exact if 0 ! Hq (U ) ! 0 implies Hq (U ) = 0: For an integer n 1; we say that U is n-acyclic if H (U ) = 0 for 1 n: Furthermore, U is acyclic if H (U ) = 0 for 1: Let F be a collection of nonempty convex sets in euclidean d-space Rd and let 0 n < d be an integer. We denote by Tn (F ) the topological space of all n-planes in Rd transversals to F ; that is, the space of nplanes that intersect all members of F , as a subset of the corresponding Grassmannian space. We say that F is separated if for every 2 n 0 d and every subfamily F F of size n, there is no (n 2)-plane 0 transversal to F : Let F = fA1 ; :::; An g be a collection of closed subsets of a metric space X and let > 0 be a real number. We denote by F = fA1 ; :::; An g the collection of open subsets of X, where A denotes [ the open n neighborhood of A X: Note that if X = R , and F is convex or [ star-shaped, respectively, then the same is true for F: 2. A Topological Helly-Type Theorem We start with the following auxiliary proposition. Proposition A(m; ) : Let F = fA1 ; :::; Am g be a family of open subsets of a topological space X and let 0 be an integer. Suppose that for 0 any subfamily F F of size j; 1 j m, \ 0 Hm 1 j+ ( F ) = 0: Then

Hm

2+

[ ( F ) = 0:

Proof. Proposition A(2;0) claims that the union of two connected sets with nonempty intersection is a connected set, and Proposition A(2; ) is just the statement of the exactness of the Mayer-Vietoris sequence: 0 = H (A1 ) H (A2 ) ! H (A1 [ A2 ) ! H 1 (A1 \ A2 ) = 0: The proof is by induction on m. In fact, we shall prove that Proposition A(m; ) together with Proposition A(m; +1) implies Proposition A(m+1; ): Suppose F = fA0 ; :::; Am g is a …nite collection of m + 1 open subsets 0 of X such for 1 j m + 1 and any subfamily F F of size j; \that [ 0 Hm j+ ( F ) = 0: We will prove that Hm 1+ ( F ) = 0: Let us …rst prove, using Proposition A(m; +1) ; that Hm 2+ (A1 [:::[Am ) = 0: This

A NEW TOPOLOGICAL HELLY THEOREM AND SOME TRANSVERSAL RESULTS3 0

is so because for any subfamily fA1 ; :::; Am g of size j; 1 j m, \ 0 F we have Hm 1 j+( +1) ( F ) = 0: Let us consider the Mayer-Vietoris exact sequence of the pair ((A1 [ ::: [ Am ); A0 ) : [ 0 = Hm 1+ (A0 ) Hm 1+ (A1 [ ::: [ Am ) ! Hm 1+ ( F ) ! Hm 2+ (A0 \ (A1 [ ::: [ Am )) = 0: Since by hypothesis, Hm 1+ (A0 ) = 0; in order to conclude the proof of Proposition A(m+1; ) we need to prove that Hm 2+ (A0 \ (A1 [ ::: [ Am )) = 0: For that purpose, let G = fB1 ; :::; Bm g be the family of open subsets of X given by Bi = A0 \ Ai , 1 i m: Note \ 0that for any sub0 family G G of size j; 1 j m, Hm 1 j+ ( G ) = 0: This is so \ 0 \ 0 because the homology group Hm 1 j+ ( G ) = Hm (j+1)+ ( F ) = 0

0; where F is the corresponding [ subfamily of F of size j + 1. Then by Proposition A(m; ) ; 0 = Hm 2+ ( G) = Hm 2+ (A0 \ (A1 [ ::: [ Am )): This completes the proof. We now give the Topological Berge’s Theorem. See [1]. Theorem B(m; ) : Let F = fA1 ; :::; Am g be a family of open subsets of a topological space 0 be an integer. Suppose that [ X and let a) Hm 2+ ( F ) = 0; b) for 1

Then

j

m

0

1 and any subfamily F \ 0 Hm 2 j+ ( F ) = 0: H

\

1(

F of size j;

F ) = 0:

Proof. The proof is by induction. Theorem B(2;0) claims that two nonempty open sets whose union is connected must have a point in common and Theorem B(2; ) is just the statement of the exactness of the Mayer-Vietoris sequence: 0 = H (A1 [ A2 ) ! H 1 (A1 \ A2 ) ! H 1 (A1 ) H 1 (A2 ) = 0: Let us prove that Theorem B(m; ) implies Theorem B(m+1; ): Let [ F = fA0; A1 ; :::; Am g as in Theorem B(m+1; ) : That is, Hm 1+ ( F ) = 0

0 and for 1\ j m and any subfamily F F of size j; we have 0 Hm 1 j+ ( F ) = 0: Let G = fB1 ; :::; Bm g; where Bi = A0 \ Ai ; \ \ 1 i m: In order to prove that H 1 ( F ) = H 1 ( G) = 0; we must show that the family G = fB1 ; :::; Bm g satis…es properties a) and b) of Theorem B(m; ) : Proof of [a). We need to prove that Hm [2+ (A0 \(A1 [:::[Am 1 )) = Hm 2+ ( G) = 0: Note that Hm 1+ ( F ) = 0 and Hm 2+ (A0 ) = 0: Furthermore, by Proposition A(m; ) ; for the family fA1 ; :::; Am g; we

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L. MONTEJANO

have that Hm 2+ (A1 [ ::: [ Am ) = 0: Thus the conclusion follows from the Mayer-Vietoris exact sequence of the pair (A0 ; (A1 [ ::: [ Am ); 0 = Hm

(A0 [ ::: [ Am ) ! Hm 2+ (A0 \ (A1 [ ::: [ Am )) ! Hm 2+ (A0 ) Hm 2+ (A1 [ ::: [ Am ) = 0:

1+

0

of b). For 1 j m 1 and any subfamily G G of \Proof \ \size0 j; 0 0 0 G = F ; where F F has size j+1: Thus Hm 1 (j+1)+ ( F ) = \ 0 Hm 2 j+ ( G ) = 0: This completes the proof of Theorem B(m+1; ) : An immediate corollary of this theorem when

= 0 is the well known

Berge’s Theorem. [2] Let F = fA1 ; :::; Am g be a family of closed convex set in Rd whose union is convex. If the intersection of every m 1 of these sets is nonempty, then their intersection is nonempty. Proof. The family F of open, convex subsets has the property that [ F is convex and the intersection of every m 1 of these open subsets \ is convex and nonempty, then since F 6= ?, for every > 0; we have \ F 6= ?: We now state our main theorem. Topological Helly Theorem. Let F be a …nite family of open subsets of a topological space X. Let d > 0 be and integer such that Hi (U ) = 0 for i d and every open subset U of X: Suppose that \ 0 Hd j ( F ) = 0 for any subfamily F

0

F of size j; 1 j \ F 6= ?:

d + 1: Then

\ Furthermore, F is acyclic. Proof. Suppose the size of F is m. Take an integer 3 n m + 1: Using Theorem B(n 1+ ; ) , from = 0 up to = n 3; we can prove the following: Claim Cn : Suppose that for every 1 j n 1 and any subfamily 0 F F of size j; [ 0 Hn 3 ( F ) = 0; and for every 1

j

Then for every 1

n

j

2 and any subfamily F \ 0 Hn j 3 ( F ) = 0:

0

F of size j;

1 and any subfamily F \ 0 Hn j 2 ( F ) = 0:

n

0

F of size j;

A NEW TOPOLOGICAL HELLY THEOREM AND SOME TRANSVERSAL RESULTS5

Assume now H (U ) = 0 for every d and every open U X and suppose that d n 3: By repeating \ the use of Claim Cn ; from n = d + 3 up to n = m + 1; we obtain that F 6= ?: Arguing as above and using Theorem B(m 1 ; ) , from = 0 up \ to = m 3; we obtain that H0 ( F ) = 0: Thus our conclusion of acyclicity can be achieved by repeating the use of Theorem B(n; ) ; 2 n m 1; 1 m 3: This concludes the proof of our main theorem. For completeness, we include here a Topological Breen’s Theorem. Theorem m : Let F = fA1 ; :::; Am g be a family of open subsets of a topological space X. Suppose that for 1 j m and any subfamily 0 F F of size j; [ 0 Hj 2 ( F ) = 0: Then

\

F 6= ?:

Proof. The proof is by induction. Theorem 2 claims that two nonempty open sets whose union is connected must have a point in common. Suppose Theorem m is true and let F = fA0 ; A1 ; :::; Am g be a family of open subsets of X such for 1 j m and any [ that 0 0 subfamily F F of size j; Hj 2 ( F ) = 0: 0

Let us prove …rst that for any subfamily F fA2 ; :::; Am g of size j; 0 j m 1; [ 0 Hj 1 ((A0 \ A1 ) [ F ) = 0:

To do so,[simply consider [ 0 the Mayer-Vietoris exact sequence of the 0 pair (A0 [ F ; A1 [ F ) : [ 0 [ 0 0 = Hj (A0 [ A1 [ F ) ! Hj 1 ((A0 \ A1 ) [ F ) ! Hj 1 (A0 [ [ 0 [ 0 F ) Hj 1 (A1 [ F ) = 0: This implies that the family fA0 \ A1 ; A2 ; :::; Am g satis…es the hypothesis of Theorem m ; and therefore by induction that A0 \ A1 \ ::: \ Am 6= ?: This completes the proof of this theorem. As an immediate consequence of the above theorem, we have the following two theorems: Topological Breen Theorem. Let F be a …nite family of open subsets of a topological space X. Let d > 0 be and integer such that H (U ) = 0 for d and every open subset U of X. Suppose that [ 0 Hj 2 ( F ) = 0; for 1

j

d + 1 and any subfamily F

0

F of size j: Then

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L. MONTEJANO

\

F 6= ?;

Breen’s Theorem. [5] Let F be a …nite family of closed convex set in Rd . If every d + 1 or fewer sets in F have a star-shaped union, then its intersection is nonempty . µ Remark. The corresponding theorems for Cech cohomology groups are also true. Furthermore, the theorems in this section are true for a class of sets A0i s in which the Mayer-Vietoris exact sequence of the pair (A0 ; (A1 [ ::: [ Am ) is exact. For example, when X is a polyhedron and every Ai X is a subpolyhedron.

3. Transversal Theorems 3.1. Preliminary Lemmas. We start with some useful lemmas concerning transversals to small families of convex sets. Lemma 3.1.1. Let A be a nonempty convex sets in Rd and let 1 n < d: Then Tn (fAg) is homotopically equivalent to G(n; d); the Grassmannian space of all n-planes in Rd through the origin. Proof. Let : Tn (fAg) ! G(n; d) be given as follows: for every H 2 Tn (fAg); let (H) be the unique n-plane through the origin parallel 1 to H. Then if 2 G(n; d); ( ) is homeomorphic to (A); where d ? : R ! is the orthogonal projection and ? 2 G(d n; d) is orthogonal to : Since has contractible …bers, then is a homotopy equivalence. Lemma 3.1.2. Let F = fA1 ; A2 ; :::; An g be a separated family of nonempty convex sets in Rd ; 2 n d; and let n m d be an m n integer. Then Tm 1 (F ) is homotopically equivalent to RP : Proof. We start by proving that Tn 1 (F ) is contractible. For this purpose let : A1 ::: An ! Tn 1 (F ) given by ((a1 ; :::; an )) be equal to the unique (n 1)-plane in Rd through fa1 ; :::; an g; for every (a1 ; :::; an ) 2 A1 ::: An : Note that is well de…ned because F is a separated family of sets. Furthermore, if H 2 Tn 1 (F ); then T 1 (H) = (H \ A1 ) ::: (H \ An ); is contractible. This implies that is a homotopy equivalence and hence that Tn 1 (F ) is contractible. Let E = f(H; ) j H is a (n 1)-plane of Rd ; is a (m 1)-plane of d R and H g: Then : E ! M (n 1; d); given by the projection in the …rst coordinate, is a classical …ber bundle with …ber RPm n , where M (n 1; d) is the Grassmannian space of all a¢ ne (n 1)-planes in Rd : Now let Y = f(H; ) 2 Tn 1 (F ) Tm 1 (F ) j H g: Clearly, the restriction j: Y ! Tn 1 (F ) is a …ber bundle with …ber RPm n and contractible base space Tn 1 (F ): Therefore j: Y ! Tn 1 (F ) is a trivial …ber bundle and hence Y is homotopically equivalent to RPm n :

A NEW TOPOLOGICAL HELLY THEOREM AND SOME TRANSVERSAL RESULTS7

Consider now the projection : Y ! Tm 1 (F ): Note that for every 2 Tm 1 (F ); the …ber 1 ( ) is equal to Tn 1 (fA1 \ ; :::; An \ g): By the …rst part of this proof, the …bers of are contractible, hence is a homotopy equivalence and Tm 1 (F ) is homotopically equivalent to RPm n : Lemma 3.1.3. Let A; B, and C be three nonempty convex sets in Rd such that A \ B = ?: Then H1 (T1 (fA; B; Cg)) = 0: Proof. Since A \ B \ C = ?; by Theorem 3 of [4], T1 (fA; B; Cg) has the homotopy type of the space of all a¢ ne con…gurations of three points in the line, achieved by transversal lines to fA; B; Cg: Note now that the space of a¢ ne con…guration of three points in a line is S1 and note further that since A \ B = ?; the space of all a¢ ne con…gurations of three points in the line achieved by transversal lines to fA; B; Cg is a subset S1 f1g; where 1 2 S1 is the a¢ ne con…guration in which the …rst and the second points coincide. This implies that H1 (T1 (fA; B; Cg)) = 0: Lemma 3.1.4. Let F = fA0 ; A1 ; A2 ; A3 g be a family of 4 closed convex set in R2 . Suppose that H0 (T1 (F )) = 0: Then there is 0 > 0 with the property that if 0 < < 0 ; then H0 (T1 (fA0 ; A1 ; A2 ; A3 g) = 0: Proof. Let A0 R2 be a closed convex set and let : M (1; 2) ! [0; 1) be the following function: for every L 2 M (1; 2); (L) is the distance from L to A0 , where M (1; 2) is the set of all lines in R2 : 1 Let us consider j T1 (fA1 ; A2 ; A3 g): Clearly, (0) = T1 (F ) which is connected. We want to proof that there is 0 > 0 with the property 1 that if 0 < < 0 ; then ([0; )) is connected. Suppose not, then 1 there is a collection of lines fLi g1 i=1 and positive real numbers fri gi=1 satisfying the following: a) fri g ! 0 and ri > ri+1 ; b) (Li ) = ri is a local minimum. That is: there is an open neighborhood Ui of Li in T1 (fA1 ; A2 ; A3 g) such that (L) ri for every L 2 Ui : Then given i 2 N; the line Li is a support line to at least two convex sets of fA1 ; A2 ; A3 g: This is so because, for example if Li is not a support line of A1 ; A2 and A3 ; then Li is not a local minimum, but the same is true if Li is not the support line of two of the Ai s: This is a contradiction because there are …nitely many support lines of two Ai s: Thus there is 0 > 0 with the property that if 0 < < 0 ; then 1 ([0; )) = T1 (fA0 ; A1 ; A2 ; A3 g) is connected. Lemma 3.1.5. Let F = fA0 ; A1 ; A2 ; A3 ) be a family of 4 pairwise disjoint, closed, smooth, convex bodies in R3 . Suppose that H0 (T1 (F )) =

8

L. MONTEJANO

0: Then there is 0 > 0 with the property that if 0 < < 0 ; then H0 (T1 (fA0 ; A1 ; A2 ; A3 g) = 0: Proof. Let A0 R3 be a smooth compact, convex body let : M (1; 3) ! [0; 1) the following function: for every L 2 M (1; 3); (L) is the distance from L to A0 , where M (1; 3) is the set of all lines in R3 : Remember that M (1; 3) is a 4-manifold with a di¤erentiable structure and that is a smooth map, see [11]. Let A1 ; A2 R3 be two smooth, disjoint, compact convex bodies. 1 2 Let (A ; A ) M (1; 3) be the set of all lines in R3 tangent to A1 and tangent to A2 : We know from [6] that (A1 ; A2 ) is a 2-dimensional torus smoothly embedded in M (1; 3) and hence that j (A1 ; A2 ) : (A1 ; A2 ) ! [0; 1) is a smooth map. 1 Let us consider j T1 (fA1 ; A2 ; A3 g): Clearly, (0) = T1 (F ) which is connected. We want to prove that there is 0 > 0 with the property 1 that if 0 < < 0 ; then ([0; )) is connected. Suppose not, then 1 there is a collection of lines fLi g1 i=1 and positive real numbers fri gi=1 satisfying the following: a) fri g ! 0 and ri > ri+1 ; b) (Li ) = ri is a local minimum. That is: there is an open neighborhood Ui of Li in T1 (fA1 ; A2 ; A3 g) such that (L) ri for every L 2 Ui : Therefore, given i 2 N; There is j 6= k 2 f1; 2; 3g such that Li 2 (Aj ; Ak ): This is so because, for example, if Li intersects the interiors of A1 ; A2 and A3 ; then Li is not a local minimum, but the same is true if Li intersects the interior of two of the Ai s: This implies that Li is a local minimum of the smooth map j (Aj ; Ak ) : (Aj ; Ak ) ! [0; 1); for some j 6= k 2 f1; 2; 3g: But hence this implies, without loss of generality, that the smooth map j (A1 ; A2 ) : (A1 ; A2 ) ! [0; 1) has in…nitely many local minimums which is impossible because this map is smooth. Thus there is 0 > 0 with the property that if 0 < < 0 ; 1 then ([0; )) = T1 (fA0 ; A1 ; A2 ; A3 g) is connected. Lemma 3.1.6. Let F = fA1 ; :::; Ad+1 ) be a separated family of closed convex sets in Rd . Suppose that H0 (Td 1 (F )) = 0: Then there is 0 > 0 with the property that if 0 < < 0 ; then H0 (Td 1 (F )) = 0: Proof. By Theorem 1 of [4], the space of transversals Td 1 (F ) of a separated family of convex sets in Rd has …nitely many components and each one of them is contractible. In fact, each component corresponds precisely to a possible order type, of d 1 points in a¢ ne (d 1)space, achieved by the transversal hyperplanes when they intersect the family F . In our case, since H0 (Td 1 (F )) = 0; we have that Td 1 (F ) is contractible and that the transversal hyperplanes intersect the family F consistently with a precise order type : Suppose now the lemma is not true, then there exist an order type ; and a collection of hyperplanes Hi that intersect 0 , di¤erent from

A NEW TOPOLOGICAL HELLY THEOREM AND SOME TRANSVERSAL RESULTS9

F i consistently with the order type 0: Since we may assume that f i g ! 0 and fHi g ! H; where H is a transversal hyperplane to F consistently with the order type 0 ; we have a contradiction. 3.2. Transversal Lines in the Plane. A family of sets is called semipairwise disjoint if, given any three elements of F , two of them are disjoint. Theorem 3.2.1. Let F be a semipairwise disjoint family of at least 6 0 compact convex sets in R2 : Suppose that for every subfamily F F 0 0 of size 5, T1 (F ) 6= and for every subfamily F F of size 4, 0 T1 (F ) is connected. Then T1 (F ) 6= ?. Proof. Let us …rst prove the theorem for a family of open, convex sets. Let X be the space of all lines in R2 : Hence H (U ) = 0 for 2 and every open subset U X. We are interested in applied the Topological Helly Theorem when d = 4. Note …rst that H3 (T1 (fAg) = 0 for every A 2 F; and H2 (T1 (fA; Bg) = 0 for A 6= B 2 F: By Lemma 3.1.3 and 0 the fact that F is semipairwise disjoint, we have that H1 (T1 (F )) = 0; 0 0 for any subfamily F F of size 3. By hypothesis, H0 (T1 (fF g) = 0; 0 0 for any subfamily F F of size 4; and H 1 (T1 (fF g) = 0; for any 0 subfamily F F of size 5. This implies, by the Topological Helly Theorem, that T1 (F ) is nonempty. Let F = fA1 ; A2 ; :::; An g and let m 1 be positive integer. By Lemma 3.1.4, there is 0 < 1 < m1 ; such that F1 = fA11 ; A2 ; :::; An g is a family of convex sets in R2 satisfying the hypothesis of Theorem 3.2.1. Similarly, there is 0 < 2 < m1 ; such that Fj = fA11 A22 ; :::; An g is a family of convex sets in R2 satisfying the hypothesis of Theorem 3.2.1. By repeating this argument we end up with a family of open convex sets Fn = fA11 A22 ; :::; Ann g satisfying the hypothesis of Theorem 3.2.1, and therefore, by the …rst part of this proof, with the property that T1 (Fn ) 6= ?: Since this is true for every positive integer m 1; the fact that the set of all transversal lines that intersect a compact set is compact implies that T1 (F ) 6= ?: 3.3. Transversal Lines in 3-Space. In this section we study transversal lines to families of convex sets in R3 : Theorem 3.3.1. Let F be a pairwise disjoint family of at least 6 0 smooth, convex bodies in R3 . Suppose that for any subfamily F F 0 0 of size 5, T1 (F ) 6= ?; and for any subfamily F F of size 4, 0 T1 (F ) is connected. Then, T1 (F ) 6= ?. Proof. Let us …rst assume that our convex sets are open, convex subsets of R3 . Let X be the space of all lines in R3 ; hence X is an open 4-dimensional manifold and therefore H (U ) = 0 for 4 and every open subset U X. We are interested in applied the Topological Helly

10

L. MONTEJANO

Theorem for d = 4. By Lemma 3.1.1, H3 (T1 (fAg) = 0; for every A 2 F; since T1 (fAg) has the homotopy type of G(1; 3) = RP2 : By Lemma 3.1.2, H2 (T1 (fA; Bg) = 0; for every A 6= B 2 F: By Lemma 3.1.3 0 and the fact that F is pairwise disjoint, we have that H1 (T1 (F )) = 0; 0 0 for any subfamily F F of size 3. By hypothesis, H0 (T1 (fF g) = 0; 0 0 for any subfamily F F of size 4; and H 1 (T1 (fF g) = 0; for any 0 subfamily F F of size 5. This implies, by the Topological Helly Theorem, that T1 (F ) is nonempty. Let F = fA1 ; A2 ; :::; An g and let m 1 be positive integer. By Lemma 3.1.5, there is 0 < 1 < m1 ; such that F1 = fA11 ; A2 ; :::; An g is a family of convex sets in R3 satisfying the hypothesis of Theorem 3.3.1. The proof follows now the same arguments that the last part of the proof of Theorem 3.2.1. Corollary 3.3.2. Let F be a pairwise disjoint family of at least 6 0 smooth, convex bodies in R3 . Suppose that for any subfamily F F 0 0 of size 5, T1 (F ) 6= ?; and for any subfamily F F of size 4, there 0 is exactly one transversal line to F . Then, F has a unique transversal line.

3.4. Transversal Hyperplanes. This section is devoted to stating and proving a theorem concerning transversal hyperplanes to families of separated convex sets in d-space. Theorem 3.4.1. Let F be a separated family of at least d + 3 closed, 0 convex sets in Rd : Suppose that for any subfamily F F of size 0 0 d + 2, Td 1 (F ) 6= ? and for any subfamily F F of size d + 1, 0 Td 1 (F ) is connected. Then Td 1 (F ) 6= ?. Proof. Let us …rst prove the theorem for a separated family of open convex sets. We are going to use the Topological Helly Theorem. Let X be the space of all hyperplanes of Rd : Note that H (U ) = 0 for d and every open subset U X: In particular, H (U ) = 0 for every d +1: 0 By Lemma 3.1.2, for every subfamily F F of size j, 1 j d; 0 0 Td 1 (F ) is\ homotopically equivalent to RPd n and hence Hd j+1 (Td 1 (F )) 0 = Hd j+1 ( fTd 1 (fAg) j A 2 F g) = 0: Furthermore, by hypothesis, the same is true for j = d + 1 and j = d + 2: Consequently, by our Topological Helly Theorem, Td 1 (F ) 6= ?. By Lemma 3.1.6, there is > 0; such that F is a separated family 0 of open, convex sets in Rd and for any subfamily F F of size d + 0 1; Td 1 (F ) is connected. By the above, this implies that Td 1 (F )) 6= ?: Hence, by completeness of the Grassmannian spaces, Td 1 (F ) 6= ?.

A NEW TOPOLOGICAL HELLY THEOREM AND SOME TRANSVERSAL RESULTS 11

References [1] I. Bárány, J. Matoušek, Berge’s theorem, fractional Helly and art galleries, Discrete Math., 106 (1994) 198-215. [2] C.Berge, Sur une propriété combinatoire des ensembles convexes, C.R. Acad. Sci. Paris, 248 (1959) 2698-2699. [3] S.A. Bogatyi, Topological Helly theorem, Fundam. Prikl. Mat., 8(2) (2002), 365-405. [4] J. Bracho, L. Montejano, D. Oliveros. The topology of the space of transversals through the space of con…gurations, Topology and its Applications, 120(1–2) (2002), 92–103. [5] M. Breen, Starshaped unions and nonempty intersections of convex sets in Rd ; Proc. AMS 108 (1990), 817-820. [6] S.E. Cappell, J.E. Goodman, J. Pach, R. Pollack, M. Sharir. Common tangent and common transversals, Advances in Mathematics, 106 (1994), 198-215 [7] L. Danzer, B. Grünbaum, and V. Klee, Helly’s theorem and its relatives. In V.Klee, editor, Convexity, Proc. of Symposia in Pure Math., pages 101-180. Amer. Math. Soc., 1963. [8] H.E. Debrunner, Helly type theorems derived from basic singular homology, The American Mathematical Montly, 17(4) (1970), 375–380. [9] J. Eckho¤, Helly, Radon and Carathéodory type theorems, in Handbook of Convex Geometry (P.M. Gruber and J.M. Wills Eds.), North-Holland, Amsterdam, 1993. [10] G.Kalai and R. Meshulam, Leray numbers of projections and a topological Helly type theorem, Journal of Topology 1 (2008), 551-556 [11] L. S. Pontryagin, Characteristic cycles on di¤ erential manifolds, Translations of the Amer. Math. Soc. No.32 (1950), 149-218. E-mail address: [email protected] Luis Montejano, Instituto de Matemáticas, Unidad Juriquilla., Universidad Nacionál Autónoma de México, Juriquilla Queretaro. 66230.