  Using a potential theory on metric graphs "Gamma", we introduce the notion of
tropical convexity to the space "RDiv^d(Gamma)" of effective R-divisors of
degree d on "Gamma" and show that a natural metric can be defined on
"RDiv^d(Gamma)". In addition, we extend the notion of reduced divisors which is
conventionally defined in a complete linear system |D| with respect to a single
point in "Gamma". In our general setting, a reduced divisor is defined uniquely
as an R-divisor in a compact tropical convex subset "T" of "RDiv^d(Gamma)" with
respect to a certain R-divisor "E" of the same degree d. In this sense, we
consider reduced divisors as canonical projections onto "T". We also
investigate some basic properties of tropical convex sets using techniques
developed from general reduced divisors.
