  A two-terminal interactive function computation problem with alternating
messages is studied within the framework of distributed block source coding
theory. For any finite number of messages, a single-letter characterization of
the sum-rate-distortion function was established in previous works using
standard information-theoretic techniques. This, however, does not provide a
satisfactory characterization of the infinite-message limit, which is a new,
unexplored dimension for asymptotic-analysis in distributed block source coding
involving potentially an infinite number of infinitesimal-rate messages. In
this paper, the infinite-message sum-rate-distortion function, viewed as a
functional of the joint source pmf and the distortion levels, is characterized
as the least element of a partially ordered family of functionals having
certain convex-geometric properties. The new characterization does not involve
evaluating the infinite-message limit of a finite-message sum-rate-distortion
expression. This characterization leads to a family of lower bounds for the
infinite-message sum-rate-distortion expression and a simple criterion to test
the optimality of any achievable infinite-message sum-rate-distortion
expression. For computing the amplewise Boolean AND function, the
infinite-message minimum sum-rates are characterized in closed analytic form.
These sum-rates are shown to be achievable using infinitely many
infinitesimal-rate messages. The new convex-geometric characterization is used
to develop an iterative algorithm for evaluating any finite-message
sumrate-distortion function. It is also used to construct the first examples
which demonstrate that for lossy source reproduction, two messages can strictly
improve the one-message Wyner-Ziv rate-distortion function settling an
unresolved question from a 1985 paper.
