  The convex and metric structures underlying probabilistic physical theories
are generally described in terms of base normed vector spaces. According to a
recent proposal, the purely geometrical features of these spaces are
appropriately represented in terms of the notion of `measure cone' and the
`mixing distance' [1], a specification of the novel concept of `direction
distance' [2]. It turns out that the base norm is one member of a whole
characteristic family of `mc-norms' from which it can be singled out by virtue
of a certain orthogonality relation. The latter is seen to be closely related
to the concept of minimal decomposition. These connections suggest a simple
geometric interpretation of the familiar notion of the disjointness of
(probability) measures and the Hahn-Jordan decomposition of measures which has
been addressed briefly in [1] and will be elaborated here. The results obtained
give an indication of the extent to which a general measure cone admits measure
theoretic interpretations.
  [1] P. Busch, E. Ruch: The Measure Cone -- Irreversibility as a Geometrical
Phenomenon, Int. J. Quant. Chem. 41 (1992) 163-185.
  [2] E. Ruch: Der Richtungsabstand}, Acta Applic. Math. 30 (1992) 67-93.
