We consider the
minimization problem of φ-divergences between a given probability measure
and subsets Ω of the vector space M
of all signed measures which integrate a given class F of bounded or
unbounded measurable functions. The vector space M
is endowed with the weak topology induced by the
class F ∪ B
the class of all bounded measurable functions. We treat the problems of
existence and characterization of the φ-projections of
We also consider the dual equality and the dual attainment problems when Ω
is defined by linear constraints.
(X) of total mass 0 and is strictly quasihypermetric if in addition the equality I(μ) = 0 holds amongst measures μ of mass 0 only for the zero measure.
This paper explores the constant M(X) and other geometric aspects of X in the case when the space X is finite, focusing first on the significance of the maximal strictly quasihypermetric subspaces of a given finite quasihypermetric
space and second on the class of finite metric spaces which are L1-embeddable. While most of the results are for finite spaces, several apply also in the general compact case. The analysis
builds upon earlier more general work of the authors  .
A technical complication is the sign of Q or μ *. Yeung ( 2008 , p. 59) noted that one has to be cautious in referring to this information measure as a signedmeasure instead of a measure (because the latter can assume only nonnegative values