 
Summary: Metric Extensions and the L 1 Hierarchy
David Avis
McGill University
Montreal, Quebec
Hiroshi Maehara
Ryukyu University
Nishihara, Okinawa
ABSTRACT
A finite semimetric is L 1  embeddable if it can be expressed as a non
negative combination of Hamming semimetrics. A finite semimetric is called
hypermetric if it satisfies the (2k + 1)  gonal inequalities which naturally gener
alize the triangle inequality. It is known that all L 1  embeddable semimetrics
are hypermetric and the metric induced by K 7  P 3 is hypermetric but not
L 1  embeddable. In the first part of the paper we show that there are infinite
metrics that are hypermetric but are not L 1  embeddable, answering a question
of M. Deza. We introduce the r  extension of a semimetric: this is the addition
of new points from which all of the distances are r. We show that all 2exten
sions of K 7  P 3 are hypermetric. Unfortunately, 2  extensions of hypermetrics
in which all of the distances are either one or two may not be hypermetric in gen
eral. We illustrate this by showing that the metric induced by the star K 1,5 (which
