Several Open Problems

Abstract

We have established in Section 6 that, given a metric space (T, ρ) can be isometrically embedded into L¹, there exists a Lévy Brownian function over (T, ρ), and moreover, one can construct an indicator model for this function. The converse is obviously true: If both a Brownian function and an indicator model for this function exist, then (T, ρ) may be isometrically embedded into L¹. However, a more natural question is the following: Does the existence of a Brownian function imply that (T, ρ) may be isometrically embedded into L¹? In the general case, the answer is in the negative. It turns out that all the metric subspaces of L¹ possess a special property, they are hypermetric; but an example may be given of a metric space with a finite number of points, with a Brownian function defined on it, which is however not hypermetric (see [Assl-Ass4]). Nevertheless, for the class of normed spaces, the existence of a Brownian function implies that the space may be embedded into L¹, and hence an indicator model exists [B—DC—K, Gag]. A similar statement is apparently true for a wider class of spaces, for instance, for the homogeneous spaces. The homogeneity may be interpreted, for example, in the same sense as it was done in Theorem 15.3, when we considered the continuity conditions. We formulate this conjecture as the following problem (in this section, we say ‘problem’ to describe a statement to be proved or refuted; certain elements of the assumptions involved may be loosely interpreted or require some additional adjustments).