Tropical Combinatorics

A = A(V ): max-tropical hyperplane arrangement in T d−1
Key Observation
Proposition
For p, q ∈ T d−1 let r ∈ tconv max {p, q} be an arbitrary point on the
max-tropical line segment between p and q. Then for arbitrary
k ∈ [d]:
tA(r)k ≤ max{tA(p)k,tA(q)k}
Corollary
For arbitrary b ∈ Nd :
(CA)≤b := p ∈ T d−1
: tA(p) ≤ b = {C ∈ CA : tA(C) ≤ b}
max-tropically convex and hence contractible.