Tropical Convexity and Canonical Projections

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.