# [Banach] Abstract of a paper by Konrad J. Swanepoel

Dale Alspach alspach at www.math.okstate.edu
Fri Sep 22 07:41:17 CDT 2006

This is an announcement for the paper "A problem of Kusner on
equilateral sets" by Konrad J. Swanepoel.

Abstract: R. B. Kusner [R. Guy, Amer. Math. Monthly 90 (1983),
196--199] asked whether a set of vectors in a d-dimensional real
vector space such that the l-p distance between any pair is 1, has
cardinality at most d+1. We show that this is true for p=4 and any
d >= 1, and false for all 1<p<2 with d sufficiently large, depending
on p.
More generally we show that the maximum cardinality is at most
$(2\lceil p/4\rceil-1)d+1$ if p is an even integer, and at least
$(1+\epsilon_p)d$ if 1<p<2, where $\epsilon_p>0$ depends on p.

Archive classification: Metric Geometry; Functional Analysis

Mathematics Subject Classification: 52C10 (Primary) 52A21, 46B20
(Secondary)

Citation: Archiv der Mathematik (Basel) 83 (2004), no. 2, 164--170

Remarks: 6 pages. Small correction to Proposition 2

The source file(s), kusner-corrected.tex: 19322 bytes, is(are)
stored in gzipped form as 0309317.gz with size 7kb. The corresponding
postcript file has gzipped size 43kb.

Submitted from: swanekj at unisa.ac.za

The paper may be downloaded from the archive by web browser from
URL

http://front.math.ucdavis.edu/math.MG/0309317

or

http://arXiv.org/abs/math.MG/0309317

or by email in unzipped form by transmitting an empty message with
subject line

uget 0309317

or in gzipped form by using subject line

get 0309317

to: math at arXiv.org.