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(a)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(a)arXiv.org.
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(a)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(a)arXiv.org.
This is an announcement for the paper "Classes of strictly singular
operators and their products" by George Androulakis, Gleb Sirotkin,
and Vladimir G. Troitsky.
Abstract: V.~D. Milman proved in~\cite{Milman:70} that the product
of two strictly singular operators on $L_p[0,1]$ ($1\le p<\infty$)
or on $C[0,1]$ is compact. In this note we utilize Schreier families
$\mathcal{S}_\xi$ in order to define the class of $\mathcal{S}_\xi
$-strictly singular operators, and then we refine the technique of
Milman to show that certain products of operators from this class
are compact, under the assumption that the underlying Banach space
has finitely many equivalence classes of Schreier-spreading sequences.
Finally we define the class of ${\mathcal S}_\xi$-hereditarily
indecomposable Banach spaces and we examine the operators on them.
Archive classification: Functional Analysis
Mathematics Subject Classification: 47B07, 47A15
The source file(s), compactproducts.tex: 76155 bytes, is(are) stored
in gzipped form as 0609039.gz with size 22kb. The corresponding
postcript file has gzipped size 102kb.
Submitted from: giorgis(a)math.sc.edu
The paper may be downloaded from the archive by web browser from
URL
http://front.math.ucdavis.edu/math.FA/0609039
or
http://arXiv.org/abs/math.FA/0609039
or by email in unzipped form by transmitting an empty message with
subject line
uget 0609039
or in gzipped form by using subject line
get 0609039
to: math(a)arXiv.org.