Tropical Combinatorics

Tropical Convexity
for x, y ∈ R d let [Zimmermann 1977] [Develin & Sturmfels 2004]
[x, y]trop := {(λ ⊙ x) ⊕ (µ ⊙ y) : λ, µ ∈ R}
S ⊆ R d tropically convex: [x, y]trop ⊆ S for all x, y ∈ S
S tropically convex ⇒ λ ⊙ S = λ1 + S ⊆ S for all λ ∈ R
consider tropically convex sets in T d−1 = R d /R1
recall: we identify
(x0,x1,...,xd) + R1 = (0,x1 − x0,...,xd − x0) + R1
with (x1 − x0,...,xd − x0)
tropical polytope := tropical convex hull
of finitely many points in T d−1 ≈ R d−1