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Equilateral sets in ln Pavel Pudlak
 

Summary: Equilateral sets in ln
p
Noga Alon
Pavel PudlŽak
May 23, 2002
Abstract
We show that for every odd integer p 1 there is an absolute positive constant cp,
so that the maximum cardinality of a set of vectors in Rn 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
cardinality of such a set in Rn
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,
the lp-distance between two points a = (a1, . . . an) and b = (b1, . . . , bn) in Rn
is a - b p =
( n
k=1 |ai - bi|p

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University

 

Collections: Mathematics