  The Kashiwara $B(\infty)$ crystal pertains to a Verma module for a Kac- Moody
Lie algebra. Ostensibly it provides only a parametrisation of the
global/canonical basis for the latter. Yet it is much more having a rich
combinatorial structure from which one may read of a parametrisation of the
corresponding basis for any integrable highest weight module, describe the
decomposition of the tensor products of highest weight modules, the Demazure
submodules of integrable highest weight modules and Demazure flags for
translates of Demazure modules. $B(\infty)$ has in general infinitely many
presentations as subsets of countably many copies of the natural numbers each
given by successive reduced decompositions of Weyl group elements. In each
presentation there is an action of Kashiwara operators determined by Kashiwara
functions. These functions are linear in the entries. Thus a natural question
is to show that in each presentation the subset $B(\infty)$ is polyhedral. Here
a new approach to this question is initiated based on constructing dual
Kashiwara functions and in this it is enough to show that the latter are also
linear in the entries.
  The present work resolves one of the two very difficult obstacles in a
step-wise construction, namely that the resulting functions must satisfy a sum,
or simply S, condition. It depends very subtly on inequalities between the
coefficients occurring in functions obtained from the previous step. The only
remaining obstacle, that sufficiently many functions are obtained, can at least
be verified in many families of cases, though this is to be postponed to a
subsequent paper.
  This theory has some intriguing numerology which involves the Catalan numbers
in two different ways.
