H. Tverberg; A generalization of Radon's theorem. J. London Math. Soc., 41, 1966, 123--128. 7  Home/Search   Document Not in Database   Summary   Related Articles   Check

....except for a finite number of discrete instances, where the sign of exactly one (d 1) tuple changes. Of course, this paradigm is not new for the analysis of configurations in combinatorial geometry. Tverberg s original proof of his famous generalization of Radon s theorem is a prominent example [Tv]. Recently, continuous motions were used also in the context of k sets by Gullikson and Hole [GH] 2 Planar identities Let us briefly recapitulate the set up for this section. We are given a set P of n points in the plane, n even, such that no three points lie on a common line. A halving edge is ....

Helge Tverberg, A generalization of Radon's Theorem, J London Math Soc 41 (1966) 123--128

....22 1 C C C C C C C C A 18 TOC Existence of Good Q Codes . Let E d be a t dimensional subspace of error operators and C a size k c code. Theorem 8: There exists an E d detecting q code S # span C of dimension # k (t 1) Proof: Based on a convex intersection argument. See [7, 6]. Corollary 9: There exists an E d detecting q code of dimension # N (t(t 1) Problem 10: What is the minimum size Q(t, N) of the largest E d detecting q code Cor. 9: Q(t, N) # N (t(t 1) 19 TOC From Error Detection to Error Correction . Error algebra: E # # . ....

H. Tverberg. A generalization of radon's theorem. J. London Math. Soc., 41:123--128, 1966.

....[10] 2.2 Tverberg s theorem Theorem 2.2 (Tverberg s Theorem) Every (d 1) r 1) 1 points in R d can be partitioned into r parts such that the convex hulls of these parts have nonempty intersection. Figure 1 about here Proofs of Tverberg s theorem were given by Tverberg ( 66) [27], Doignon and Valette ( 77) Tverberg ( 81) Tverberg and Vrecica ( 92) Sarkaria ( 92) 25] and Roudne ( 99) 23] While the original proof was quite dicult the proofs of Sarkaria and Roudne are remarkably simple. Roudne s recent proof is by minimizing the sum of squares of the r distances ....

H. Tverberg, A generalization of Radon's Theorem, J. London Math. Soc. 41 (1966), 123-128.

....the upper bound (which corresponds to the standard symmetric decomposition of the ball to n 1 regions, is the truth. Perhaps also here the natural conjecture is false 1. 2 Helly type theorems Tverberg s theorem Sarkaria [40] found a striking simple proof of the following theorem of Tverberg: [49] Every (d 1) r Gamma 1) 1 points in R d can be partitioned into r parts such that the convex hulls of these parts have nonempty intersection. He used the following result of Barany [2] Let A 1 ; A 2 ; A d 1 be sets in R d such that x 2 conv(A i ) for every i. Then it is ....

H. Tverberg, A generalization of Radon's Theorem, J. London Math. Soc. 41 (1966), 123-128.

....sites into subsets, the convex hulls of which all have a common intersection. The Tverberg depth of a point t is the maximum cardinality of any Tverberg partition for which the common intersection contains t. Note that the Tverberg depth is a lower bound on the location depth. Tverberg s theorem [18,19] is that there always exists a point with Tverberg depth #n (d 1)# (a Tverberg point) this result generalizes both the existence of center points (since any Tverberg point must be a center point) and Radon s theorem [12] that any d 2 points have a Tverberg partition into two subsets. Another ....

H. Tverberg. A generalization of Radon's theorem. J. London Math. Soc. 41:123--128, 1966.

....1) d partitions of S into q disjoint subsets, S = oe 1 [ Delta Delta Delta [ oe q , with conv(oe 1 ) Delta Delta Delta conv(oe q ) nonempty. We remark that even the existence of one such partition is not obvious : it was conjectured by Birch [2] in 1958 and confirmed by Tverberg [13] in 1966 only by means of a fairly involved argument (but see also remark (e) of x5) Henceforth we will refer to partitions of the above kind as Tverberg partitions of S. The proof of the above theorem is given in x4 and depends on Lemma 1 of x2 which verifies that the Euler number of a certain ....

H.Tverberg, A generalization of Radon's theorem, Jour. Lond. Math. Soc. 41 (1966), 123-128.

....the other hand, since not all ff i are 0, at least one u j is expressed as a nontrivial linear combination of points of A. This proves Proposition 2.3. 2 Bibliography and Remarks. The Colored Carath eodory Theorem is due to B ar any [B ar82] Tverberg s Theorem was proved by Tverberg (really ) Tve66] His original proof was quite complicated. The idea is simple, though: start with some point configuration for which the theorems is valid, and convert it to a given configuration by moving one point at a time. During the movement, the current partition may stop working at some point, and it ....

H. Tverberg. A generalization of Radon's theorem. J. London Math. Soc., 41:123-- 128, 1966.

....space coincide with those defined in the beginning of this section. There are several papers discussing convexity spaces and their Radon number and Helly, Caratheodory, and exchange numbers (to be defined in x3) e.g. Danzer, Grunbaum and Klee [5] Eckhoff [7] Hammer [11] Levi [14] Tverberg [18]. In particular, the Helly number of our space (Z d ; C d ) had been studied before by Doignon [6] Also, some recent investigations motivated by integer programming were concerned with Helly properties of the integer lattice (Bell [1] Scarf [15] Schrijver [16, p. 234] and Caratheodory ....

H. Tverberg, A generalization of Radon's theorem, J. London Math. Society, 41 (1966), pp. 123-128.

.... C(I j ) stands for f 2 R d ; depth( I j ) 1g: Note that Conjecture 2 would imply Conjecture 1, because a point in k j=1 C(I j ) must satisfy depth( H n ) depth( k [ j=1 C(I j ) k X j=1 depth( C(I j ) k: In fact, Conjecture 2 has the same form as the theorem of Tverberg (1966) for a configuration of n points in R d , if we read C(I j ) as the convex hull instead of the contractible hull. Tverberg used this result to give an alternative proof the Neumann Rado theorem about the existence of a point with location depth k. Up to now we have considered arbitrary ....

Tverberg, H. (1966), A generalization of Radon's Theorem, Journal of the London Mathematical Society, 41, 123-128.

....I 2 = and we are done. 7 Tverberg Colourings A Tverberg k colouring of a set S of points in IR d is a k colouring of S such that the convex hulls of the monochromatic subsets have a point in common, i.e. a partition S = U k i=1 S i such that the intersection T k i=1 conv(S i ) is nonempty. Tverberg (1966) proved the following, so called Tverberg s theorem. Theorem 7.1 Any set of more than (k Gamma 1) d 1) points in IR d has a Tverberg k colouring. This theorem will follow from the proof of Theorem 7.2 below. Note that for k = 2, it reduces to the fact that any set of more than d 1 points in ....

Tverberg H. (1966). A generalization of Radon's theorem. J. London Math. Soc. 41 123--128.

....; S(I k ) have a nonempty intersection. This would imply Conjecture 1 because for any point in that intersection rdepth( Z n ) k. If we were to replace hyperplanes by points, and S(I j ) by the convex hull of I j , then Theorem 9 and Conjecture 2 correspond to results of Birch (1959) and Tverberg (1966). 6 Relation with location depth The general definition of depth (Definition 1) can be applied to the multivariate location setting as well. The data set X n then consists of n observations x i 2 R p , and a candidate fit is itself a p variate point which should describe the position of the ....

Tverberg, H. (1966), "A Generalization of Radon's Theorem," Journal of the London Mathematical Society, 41, 123-128.

....except for a finite number of discrete instances, where the sign of exactly one (d 1) tuple changes. Of course, this paradigm is not new for the analysis of configurations in combinatorial geometry. Tverberg s original proof of his famous generalization of Radon s theorem is a prominent example [Tv]. Recently, continuous motions were used also in the context of k sets by Gullikson and Hole [GH] 2 Planar identities Let us briefly recapitulate the set up for this section. We are given a set P of n points in the plane, n even, such that no three points lie on a common line. A halving edge is ....

Helge Tverberg, A generalization of Radon's Theorem, J London Math Soc 41 (1966) 123--128

....of Corollary 1: No line crosses more than n 2 =8 halving triangles. While the bound in R 3 still allows for a simple proof, the situation gets more involved in dimensions 4 and higher, where the best bounds due to P. Agarwal et al. AACS] are based on a colored version of Tverberg s Theorem [Tv] by R. T. Zivaljevi c and S. T. Vre cica [ZV] 2 The general bound of O(max(n; 3 p cn 2 ) in the crossing lemma [ACNS, Le] is obtained with x = min(1; 3 p n=c) The best known constant in the asymptotic bound can be found in [PT] 3 This side is unique, since contains a point, and jP ....

Helge Tverberg, A generalization of Radon's Theorem, J London Math Soc 41 (1966) 123--128

No context found.

H. Tverberg; A generalization of Radon's theorem. J. London Math. Soc., 41, 1966, 123--128. 7

No context found.

H. Tverberg, A generalization of Radon's theorem, J. London Math. Soc. 41 (1966), 123-128.

No context found.

H. Tverberg; A generalization of Radon's theorem. J. London Math. Soc., 41, 1966, 123--128.

No context found.

H. Tverberg. A generalization of Radon's theorem. J. London Math. Soc., 41:123-128, 1966.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC