A characterization of light open maps of Euclidean spaces into Euclidean spaces

A Characterization of Euclidean Spaces

The purpose of this paper is to give an elementary proof of the fact that a Banach space in which there exist projection transformations of norm one on every two-dimensional linear subspace is a euclidean space. S. Kakutani [ l ] has pointed out that a modification of a proof due to Blaschke [2] will prove this theorem. F. Bohnenblust has been able to establish this theorem for the complex case...

Windows into non-Euclidean spaces

A Characterization of Locally Euclidean Spaces

Self-Organizing Maps on non-euclidean Spaces

We propose a new type of Self-Organizing Map (SOM) that is based on discretizations of curved, non-euclidean spaces. As an introductory example, we brieey discuss \spherical SOMs" on tesselations of the sphere for the display of directional data. We then describe the construction of \hyperbolic SOMs", using regular tesselations of the hyperbolic plane, which is a non-euclidean space characteriz...

Embedding Products of Graphs into Euclidean Spaces

For any collection of graphs G1, . . . , GN we find the minimal dimension d such that the product G1 × · · · ×GN is embeddable into R . In particular, we prove that (K5) and (K3,3) are not embeddable into R, where K5 and K3,3 are the Kuratowski graphs. This is a solution of a problem of Menger from 1929. The idea of the proof is the reduction to a problem from so-called Ramsey link theory: we s...