1.1. TROPICAL HYPERSURFACES 7 ˜ S m = 2 m = 1/2 m = −1 −2 0 1 Figure 7. The left-hand figure is ˜ S . The right-hand picture shows ˇ on the x-axis and the graph of ˇ. (1) Δ = σ∈P σ. (2) If σ ∈ P and τ ⊆ σ is a face, then τ ∈ P. (3) If σ1,σ2 ∈ P, then σ1 ∩ σ2 is a face of both σ1 and σ2. For a polyhedral decomposition P, denote by Pmax the subset of maximal cells of P. We denote by P[k] the set of k-dimensional cells of P. Indeed, to get a polyhedral decomposition P of ΔS, we just take P to be the set of images under the projection NR × R → NR of proper faces of ˜ S . A polyhedral decomposition of ΔS obtained in this way from the graph of a convex piecewise linear function is called a regular decomposition and these decompositions play an important role in the combinatorics of convex polyhedra, see e.g., [32]. We can now define the discrete Legendre transform of the triple (ΔS, P, ϕ): Definition 1.4. The discrete Legendre transform of (ΔS, P, ϕ) is the triple (MR, ˇ, ˇ) where: (1) ˇ = {ˇ | τ ∈ P} with ˇ = m ∈ MR ∃a ∈ R such that −m, n + a ≤ ϕ(n) for all n ∈ ΔS, with equality for n ∈ τ . (2) ˇ(m) = max{a | −m, n + a ≤ ϕ(n) for all n ∈ ΔS}. Let us explain this in a bit more detail. First, if σ ∈ Pmax, let mσ ∈ M be the slope of ϕ|σ. Then in fact ˇ = {−mσ}, as follows from the convexity of ϕ. Second, the formula in (2) is a fairly standard way of describing the Legendre transformed function ˇ. We think of ˇ(m) as obtained by taking the graph in NR × R of a linear function on NR with slope −m and moving it up or down until it becomes a supporting hyperplane for ˜ S . The value of this aﬃne linear function at 0 is then ˇ(m) see Figure 7. Note that if m ∈ Int(ˇ), τ then the graph of −m, · + ˇ(m) is then a supporting hyperplane for the face of ˜ S projecting isomorphically to τ.

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