Tropical Combinatorics
Tropical Combinatorics
Tropical Combinatorics
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P, Q : polytopes in R d<br />
P + Q = {p + q: p ∈ P, q ∈ Q} Minkowski sum<br />
Mixed Subdivisions<br />
Minkowski cell of P + Q = full-dimensional subpolytope<br />
which is Minkowski sum of subpolytopes of P and Q<br />
Definition<br />
Polytopal subdivision of P + Q into Minkowski cells is mixed if for<br />
any two of its cells P ′ + Q ′ and P ′′ + Q ′′ the intersections P ′ ∩ P ′′<br />
and Q ′ ∩ Q ′′ both are faces.<br />
can be generalized to finitely many summands<br />
fine = cannot be refined (as a mixed subdivision!)