Multimedians in metric and normed spaces. (English) Zbl 0790.46008

Verheul, E. R.

CWI Tracts. 91. Amsterdam: CWI. x, 136 p. Dfl. 40.00 /sc (1993).

The main aim of this monograph is to give an introduction to the subject of median algebras (it is very important with respect for instance for the discrete mathematics, topology and lattice theory). The author presents a study of the modular spaces (see I, §4), in which the median algebras play the main role and moreover considers some applications of those algebras.
Part 1 contains basic definitions, results and remarks on median spaces, multilattices, modular spaces, median interval spaces. Next, the author very precisely presents the theory of modular metric spaces (part II), modular normed spaces (part III) and amalgamating spaces (part IV). The main result of part IV states that each modular Banach space can be decomposed into a rigid space and an \(\ell_ 1(I)\) space. Moreover, in this monograph we have some remarks on isometric embeddings of median spaces (part V) and on medians versus Steiner trees (part VI). This monograph is very interesting and very valuable and has a long list of papers and books.

Reviewer: A.Waszak (Poznań)

MSC:

46B20	Geometry and structure of normed linear spaces
06C05	Modular lattices, Desarguesian lattices
46A80	Modular spaces
46B25	Classical Banach spaces in the general theory
52A01	Axiomatic and generalized convexity

Keywords:

median algebras; modular spaces; multilattices; median interval spaces; modular metric spaces; modular normed spaces; amalgamating spaces; isometric embeddings of median spaces; medians versus Steiner trees