Let \( w=-3x+2y \).

Let \( \mathbf{v} = (3, 5, 4, 8, 3, 0) \) and \( \mathbf{h} = (-1, 2, 0, 1, 0, 0 ) \)

Define \( P_{i} = P_{A}(\mathbf{v} + i \mathbf{h}) \) to be the 2-dimensional alcoved polytope with right hand side \( \mathbf{v} + i \mathbf{h} \).

Observe that even though \( P_{1} \) and \(P_{2} \) are Minkowski convex combinations of \( P_{0} \) and \(P_{3} \) and equivalently correspond to convex combinations of the right hand side \( \mathbf{b} \) vectors, the coefficients of \( h_{P_{i}, w}^{*} \) switched sign, even though \( h_{P_{0}, w}^{*} \) and \( h_{P_{3}, w}^{*} \) initially had all positive coefficients. We computed that:

• \( h^*_{P_0,w}= 10z^3 + 65z^2 + 25z \)
• \( h^*_{P_1, w}=-10z^3 – 37z^2 – 7z \)
• \( h^*_{P_2, w}=-9z^3 – 39z^2 – 10z \)
• \( h^*_{P_3, w}=15z^3 + 67z^2 + 18z \)

The green line represents the linear weight w = 2y – 3x.