green_57

Is it true that for every finite abelian group $G$ the convex hulls $\Phi(G)$ and $\Phi'(G)$, obtained from kernels $\phi(g) = \mathbb{E}_{x_1 + x_2 + x_3 = g} f_1(x_2, x_3) f_2(x_1, x_3) f_3(x_1, x_2)$ with $\lVert f_i \rVert_\infty \le 1$, still coincide when the third kernel is required to depend only on $x_1 + x_2$?

theorem green_57 :
  answer(sorry) ↔
    ∀ (G : Type) [AddCommGroup G] [Fintype G] [DecidableEq G],
      Φ G = Φ' G

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