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Equilateral Sets in Lp n Equilateral sets in lp Noga Alon ∗ Pavel Pudl´ak y May 23, 2002 Abstract We show that for every odd integer p 1 there is an absolute positive constant cp, ≥ n so that the maximum cardinality of a set of vectors in R such that the lp distance between any pair is precisely 1, is at most cpn log n. We prove some upper bounds for other lp norms as well. 1 Introduction An equilateral set (or a simplex) in a metric space, is a set A, so that the distance between any pair of distinct members of A is b, where b = 0 is a constant. Trivially, the maximum n 6 cardinality of such a set in R with respect to the (usual) l2-norm is n + 1. Somewhat surprisingly, the situation is far more complicated for the other lp norms. For a finite p > 1, ~ n ~ the lp-distance between two points ~a = (a1; : : : an) and b = (b1; : : : ; bn) in R is ~a b p = k − k n p 1=p ~ ~ ( k=1 ai bi ) . The l -distance between ~a and b is ~a b = max1 k n ai bi : For 1 1 ≤ ≤ P j − j n n k − k n j − j all p [1; ], lp denotes the space R with the lp-distance. Let e(lp ) denote the maximum 2 1 n possible cardinality of an equilateral set in lp . n Petty [6] proved that the maximum possible cardinality of an equilateral set in lp is at most 2n and that equality holds only in ln , as shown by the set of all 2n vectors with 0; 1- 1 coordinates. The set of standard basis vectors together with an appropriate multiple of the ∗Institute for Advanced Study, Princeton, NJ 08540, USA and Department of Mathematics, Tel Aviv University, Tel Aviv 69978, Israel. E-mail: [email protected]. Research supported in part by a State of New Jersey grant, by a USA Israeli BSF grant, by a grant from the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. yMathematical Institute, Academy of Sciences of the Czech Republic, Prague, Czech Republic. Email: [email protected]. Research supported in part by grant No. A1019901 of the Academy of Sciences of the Czech Republic and by project No. LN00A056 of the Ministry of Education of the Czech Republic. Part of this research was done while the author was visiting IAS, Princeton. 1 all 1 vector shows that e(ln) n + 1 for all 1 p < , and the set of standard basis vectors p ≥ ≤ 1 and their negatives shows that e(ln) 2n. 1 ≥ Kusner conjectured that both examples are extremal. n Conjecture 1.1 (Kusner, [3]) e(l1 ) = 2n. Conjecture 1.2 (Kusner, [3]) For every 1 < p < , e(ln) = n + 1. 1 p The assertion of Conjecture 1.1 is easy for n 2, and has been proved for n = 3 in [2] and for ≤ n = 4 in [5]. For large n, the best known upper bound is 2n 1, and the existing techniques − supplied no nontrivial upper bound. Our main result here is a significant improvement of the trivial estimate. Theorem 1.3 There exists an absolute constant c > 0 such that the cardinality of any equilateral set of vectors in ln is at most cn log n, that is: e(ln) cn log n: 1 1 ≤ The assertion of Conjecture 1.2 is easy for p = 2, is proved in [7] for all real p sufficiently close to 2 by a continuity argument, and has also been proved by Swanepoel (c.f., [7]) for n p = 4. Galvin (c.f., [7]) has shown that for every even integer p 2, e(lp ) 1 + (p 1)n. n (p+1)=(p 1) ≥ ≤ − For general p = 1 it is proved in [7] that e(l ) c(p)n − . 6 p ≤ The proof of Theorem 1.3 can be extended, with some additional effort, to provide a n similar bound for e(lp ) for every odd integer p. Theorem 1.4 For every odd integer p 1 there exists an absolute constant cp > 0 such ≥ n n that the cardinality of any equilateral set of vectors in l is at most cpn log n, that is: e(l ) p p ≤ cpn log n: Combining our basic approach here with the technique of [7] we can slightly improve the known estimates for general non-integral p as well and prove the following. Theorem 1.5 For every p 1 there exists an absolute constant c = c(p) > 0 such that n (2p+2)=(2p 1) ≥ e(l ) cn − . p ≤ Our proofs combine probabilistic, combinatorial and linear algebra tools with some results from approximation theory. The proof of Theorem 1.3 is presented in Section 2. We then describe, in Section 3, how to extend it and prove Theorem 1.4. Section 4 contains the short proof of Theorem 1.5. The final Section 5 contains some concluding remarks. 2 n 2 Equilateral sets in l1 The basic approach n In this section we prove Theorem 1.3. Let A be a set of vectors in l1 such that the distance between any two is 1. Let m = A . Consider the m m matrix M indexed by the vectors ~ j j × of A in which M ~ = ~a b 1. By subtracting this matrix from the matrix of 1's we get ~a;b k − k the identity matrix I that has full rank m. Using the vectors ~a A we shall construct an 2 approximation of M, hence also of I. The precision of approximation and the rank of the approximating matrix will depend on m. To get an upper bound on m we shall use the following well-known lemma, whose short proof is included, for the sake of completeness. Lemma 2.1 For any real symmetric matrix M, 2 ( Mi;i) rank M Pi : (1) ≥ M 2 Pi;j i;j Proof. Put r = rank M and let λ1; λ2; : : : ; λr be all nonzero eigenvalues of M. Then r r 2 2 2 λk = trace(M) = Mi;i and λ = trace(M ) = M . Therefore, by Pk=1 Pi Pk=1 k Pi;j i;j Cauchy Schwartz r r 2 2 ( λk) ( Mi;i) M 2 = λ2 Pk=1 = Pi ; X i;j X k ≥ r r i;j k=1 implying the desired result. As shown in [1], this lemma can be used for bounding the minimum possible dimension of an Euclidean space in which one can embed an m-simplex with low distortion, and it is therefore not surprising that it can be useful for our purpose here as well. In our approximation of the identity, the diagonal elements will be close to 1, hence 2 2 2 ( Mi;i) will be close to m , while M will be O(m). Thus rank M will be essentially Pi Pi;j i;j an upper bound on m. Hence to get a bound on the number of vectors we have to get a good upper bound on the rank of M. We shall construct M in such a way that rank M will be a function N(n; m), thus the bound will be the maximum m such that N(n; m) is greater or equal to the bound of Lemma 2.1. The approximation matrix will be a matrix of scalar products of vectors in a space of dimension N(n; m).1 It will be constructed by splitting each coordinate of the n coordinates of l1 into several ones and by defining for each vector ~a A a corresponding vector a~ in dimension N(n; m). Here are more details. 2 ∗ 1This is a slight simplification of what we actually will do. In our construction the dimension will depend on the particular set A, so think of N(n; m) as an upper bound on it. 3 Let us denote by δ(x) the following function 1 if x > 0 δ(x) = 0 otherwise We shall use the following equality x y = x + y 2δ(xy) min( x ; y ): (2) j − j j j j j − j j j j n We decompose the matrix of distances in l1 as follows ~a ~b = a + b 2δ(a b ) min( a ; b ): 1 X i X i i i X i i k − k i j j i j j − i j j j j The matrix ( ai + bi ) ~ has rank at most 2, hence the essential part is the matrix Pi j j Pi j j ~a;b 2δ(aibi)( min( ai ; bi )) ~. Our goal can now be stated as follows. We want to assign a Pi j j j j ~a;b vector a~ to every vector ~a A in order to approximate δ(aibi) min(ai; bi) by the scalar ∗ 2 i ~ 2 P product (a~∗; b∗). The aim is to get a good approximation of the identity matrix I = J ai + bi ! + 2 δ(aibi) min( ai ; bi )! ; − X j j X j j X j j j j i i ~a;~b i ~a;~b where J is the m m matrix of 1's, by the matrix × ~ I∗ = J ai + bi ! + 2 (a~∗; b∗) : − X j j X j j ~a;~b i i ~a;~b Then we want to use the inequality (1) as stated above, which means that the diagonal elements of I∗ should differ from 1 by a constant less than 1, say 2/3, and the sum of the squares of all (or all off-diagonal) elements should be O(m).
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