The Geometry of Metric and Linear Spaces: Proceedings of a Conference Held at Michigan State University, East Lansing, June 17–19, 1974

252 Pages·1975·2.286 MB·English

by  Leonard M. Blumenthal (auth.), Prof. L. M. Kelly (eds.)

Lecture Notes ni
Mathematics
Edited by .A Dold and .B Eckmann
490
The Geometry of Metric
and Linear Spaces
Proceedings of a Conference Held ta Michigan State
University, East Lansing, June 17-19, 1974
Edited by .L .M Kelly
galreV-regnirpS
Berlin. grebledieH (cid:12)9 New York 1975
Editor
Prof. .L M. Kelly
Department of Mathematics
Michigan State University
East Lansing
Michigan 48824/USA
Library of Congress Cataloging in Publication Data
Main entry under title:
The geometry of metric and linear spaces.
(Lecture notes in mathematics ; $90)
Bibliography: p.
Includes index.
.i Normed linear spaces--Congresses. 2. Inner
product spaces--Congresses. 3. Metric spaces--
Congresses. 4. Convex sets--Congresses.
5. Geometry--Congresses. I. Kelley, Leroy Milton,
191~- II. Series: Lecture notes in m&r~e-
maties (Berlin) ; 490.
qA3.L28 no. 490 QA322.2 510'.8s 515'.73
75-33966
AMS Subject Classifications (1971): 50-02, 50A05, 50A10, 50C05,
52A05,52A20, 52 A25, 52A35, 52 A45, 46 B10,46 C05.
ISBN 3-540-07417-1 Springer-Verlag Berlin (cid:12)9 Heidelberg (cid:12)9 NewYork
ISBN 0-387-07417-1 Springer-Verlag New York (cid:12)9 Heidelberg (cid:12)9 Berlin
This work is subject to copyright. All rights are reserved, whether the whole
or part of the material is concerned, specifically those of translation,
reprinting, re-use of illustrations, broadcasting, reproduction by photo-
copying machine or similar means, dna storage ni data banks.
Under w 54 of the German Copyright Law where copies are made for other
than private use, a fee is payable to the publisher, the amount of the fee to
be determined by agreement with the publisher.
(cid:14)9 by Springer-Verlag Berlin - Heidelberg 1975
Printed in Germany
Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
PREFACE
One of the purposes for organizing this conference was to give
formal recognition to the contributions of Professor Leonard Mascot
Blumenthal to the foundational study of the geometry of metric
spaces. The participants presented Professor Blumenthal with a scroll
dedicating the conference to his pioneering work in the field. These
proceedings are likewise dedicated to him.
The papers in this volume are essentially elaborations of those
presented to the conference either orally or by title. In one or two
instances they are notes inspired by such papers.
The papers presented seemed to fall naturally into three cate-
gories, )a( those concerned with purely metric questions, )b( those
concerned with metric geometry in general linear spaces, and )c(
those concerned with the geometry of Minkowski spaces or convexity.
We have organized the papers here in that order.
Finally we present a list of problems suggested by the parti-
cipants and edited by R.C. Guy.
Table of Contents
.i Four Point Froperties and Norm Postulates
L.M. Blumenthal
.2 On the Equilateral Feeble Four-Point Property 14
L.M. Kelly
.3 Hypermetric Spaces 17
John B. Kelly
4. Metric Problems in Elliptic Geometry 32
J.J. Seidel
5. Discrepancy and Sums of Distances Between Foints of a
Metric Space 44
Kenneth B. Stolarsky
6. Metric Embedding Techniques Applied to Geometric
Inequalities 57
Ralph Alexander
7. Angles in Metric Spaces 66
J.E. Valentine
8. Geometric Fixed Point Theory and Inwardness Conditions 74
James Caristi and W.A. Kirk
9. On Some Aspects of Fixed Point Theory in Banach Spaces 84
Michael Edelstein
i0. Mimicry in Normed Spaces 91
M.M. Day
ii. Smoothness Classification of Reflexive Spaces I 107
K. Sundaresan
12. Norm Identities which Characterize Inner Product Spaces ll6
John Oman
13. Ratio-Sequences of Chains in Connected Metric Spaces 134
Victor Klee
14. Folygons 147
Branko Grnbaum
15. k-Equilateral (2k + l-Gons Span Only Even-Dimensional Spaces 185
Jim Lawrence
lV
16 Covering Space with Convex Bodies 187
G.D. Chakerian
17 Intersectional Configurations of Convex Sets 194
C.M. Petty
18 Metric Dependence and a Sum of Distances 206
Dorothy Wolfe
19 Tverberg-Type Theorems in Convex Product Structures 212
Gerald Thompson and William R. Hare
20 Intersecting Families of Convex Cover Order Two 218
Marilyn Breen
21 A Helly-Type Theorem for Widths 227
G.T. Sallee
22 Problems 233
Richard K. Guy
CONFERENCE ON THE GEOMETRY OF METRIC AND LINEAR SPACES
PARTICIPANTS
Professor .J Adney Professor Richard Guy
Michigan State University University of Calgary
East Lansing, MI 48824 Calgary (44) Canada
Professor J.R. Alexander, Jr. Professor william Hare
University of Illinois Clemson University
Urbana, IL 61801 Clemson, SC 29631
Professor E.Z. Andalafte Professor David Kay
University of Missouri at S.L. University of Oklahoma
St. Louis, MO 63121 Normal, OK 73069
Professor David Barnette Professor John Kelly
UCD-Davis Arizona State University
Davis, CA 95616 Tempe, AZ 85281
Professor L.M. Blumenthal Professor L.M. Kelly
University of Missouri Michigan State University
Columbia, MO 65201 East Lansing, MI 48824
Professor W. Bonnice Professor J. Kinney
University of New Hampshire Michigan State University
Durham, NH 03824 East Lansing, MI 48824
Dr. Marilyn Breen Professor W.A. Kirk
University of Oklahoma University of Iowa
Norman, Oklahoma 73069 Iowa city, Iowa 52240
Professor G.D. Chakerian Mr. Murray Klamkin
U.C.D. Ford Motor Science Lab.
Davis, California 95616 POB 2053
Dearborn, MI 48121
Professor James Chew
Michigan State University Professor V. Klee
East Lansing, MI 48824 University of Washington
Seattle, WA 98195
Professor M.M. Day
University of Illinois Dr. J.B. Kruskal
Urbana, IL 61801 Bell Telephone Labs
Murray Hill, NJ 07974
Professor M. Edelstein
Dalhousie University Professor .G Ludden
Halifax, Nova Scotia Michigan State University
Canada East Lansing, MI 48824
Professor J.S. Frame Professor John Oman
Michigan State University Wisconsin State University
East Lansing, MI 48824 Oshkosh, WI 54901
Professor R. Freese Professor C.N. Petty
St. Louis University University of Missouri
St. Louis, MO 63103 Columbia, MO 65201
Professor\Branko Grunbaum Professor J.R. Reay
University of Washington Western Washington St.
Seattle, WA 98195 College
Bellingham, WA 98225
IIIV
Professor G.T. Sallee
UCD, Davis
Davis, CA 95616
Professor J.J. Seidel
Thomas J. Watson Research Center
P.O. Box 218
Yorktown Hts., NY 10598
Professor Andrew Sobczyk
Clemson University
Clemson, SC 29631
Professor S.V. Sreedharan
Michigan State University
East Lansing, MI 48824
Dean W.L. Stamey
College of Arts and Science
Eisenhower Hall
Kansas State University
Manhattan, KS 66502
Professor K.B. Stolarsky
University of Illinois
Urbana, IL 61801
Professor K. Sundaresan
University of Pittsburgh
Pittsburgh, PA 15213
Professor Joseph Valentine
Utah State University
Logan, UT 84321
Professor Dorothy Wolfe
245 Hathaway Lane
Wynnewood, PA 19096
Michigan State University - East Lansing~ Michigan
Department of Mathematics
Conference: The Geometry of Metric and Linear Spaces
Program
Monday, June 17 - Room B-IO2 Wells Hall
A.M. 9:15 Professor Blumenthal, A review of some
L.M. -
University of Missouri metric characteri-
zations of euclidean
space.
10:15-10:45 Recess and Refreshments
iO:45-11:30 Professor W.A. Kirk, - Geometric fixed point
University of Iowa theory and inwardness
conditions.
ii:30-12:15 Professor J.J. Seidel, The state of affairs
-
Technological University in elliptic space.
Eindhoven
P.M. 2:15- 3:15 Professor John B. Kelly~ - Hypermetric spaces.
Arizona State University
3:15- 3:45 Recess and Refreshments
3:45- 4:30 Professor K.B. Stolarsky, - Discrepancy and sums
University of Illinois of distance between
points of metric
space.
4:30- 5:00 Professor J. Valentine, - Angles in metric
Utah State University spaces.
P.M. 8:00- 9:00 SPECIAL LECTURE - ROOM B-IO2 Wells Hall
Dr. J.B. Kruskal, Bell Telephone Labs
"Multidimensional scaling and clustering:
Embedding of noisy data in metric spaces."
9:00- Reception V.G. Grove Mathematics Library
D-IoI Wells Hall
Tuesday, June 18 - Room B-IO2 Wells Hall
A.M. 9:15-10:15 Professor M.M. Day - Mimicry in normed
University of Illinois spaces.
iO:15-iO:30 Recess and Refreshments
10:30-11:15 Professor M. Edelstein, - Fixed point theory
Dalhousie University in linear spaces.
11:15-12:00 Professor J. Oman, - Characterization of
Wisconsin State Hilbert space.
University
P.M. 2:00-3:00 Professor V. Klee, Convex polytopes and
University of Washington mathematical program-
ming.
3:O0-3:15 Recess and Refreshments
3:15-4:00 Professor David Barnette, - Generalized Combina-
University of California torial Spheres and
Davis Facet Splitting.
P.M. 7:00 DINNER - University Club
Wednesday, June 19 - Room A-304 Wells Hall
A.M. 9:15-10:15 Professor B. Grunbaum, - Polygons
University of Washington
10:15-10:45 Recess and Refreshments
10:45-11:30 Professor C.N. Petty, Intersectional con-
University of Missouri figurations of convex
sets.
ii:30-12:15 Professor Dorothy Wolfe, Metric dependence:
Widener College Some geometric con-
sequences.
P.M. 2:00-2:45 Professor G.D. Chakerian~ - Covering Spaces with
U.C.D. - Davis Convex Bodies.
2:45-3:30 Professor K. Sundaresan, Smoothness classifi-
University of Fittsburgh cation of reflexive
spaces.
3:30-4:00 Professor G.T. Sallee - A Helly Type Theorem
for widths.
FOUR-POINT PROPERTIES AND NORM POSTULATES
Leonard M. Blumenthal
University of Missouri
i. Introduction
This paper exhibits and exploits for the frist time the re-
lations between two problems that arose about forty years ago,
attracted numerous investigators, and continue to be studied, most-
ly without reference to their inter-connections.
Those problems are (i) to determine conditions on the six
mutual distances of four points of a complete, metrically convex
ahd externally convex metric space (denoted here by ~) that are
necessary and sufficient in order that the metric of ~ may be
euclidean, and )2( to ascertain conditions on the norm of a real
Banach space (denoted here by ~) that are necessary and suffi-
cient to permit defining an inner product in ~, which is connect-
ed with the norm in the usual manner. We refer to the desired
conditions on the six mutual distances of four points of ~ as
four-point properties, and call the sought-for conditions on the
norm of ~, norm postulates. One four-point property was estab-
lished by W.A. Wilson in 1932, and a norm postulate follows from
Fr6chet's paper of 1935, and by the well-known result of Jordan and
von Neumann of the same date (explicit references will be provided
later).
The connection between the two problems stated above is made
clear by noting that )a( every Banach space S becomes a complete,
metrically convex and externally convex metric space ~ upon de-
fining the distance xy of any two elements (points) x,y by
xy = xll - YH, (b) norm postulates on E may be interpreted as
conditions on the mutual distances of the points of certain finite
subsets of ~ (quadruples of points in those cases we shall