Proposition 12 ([Initial supporting triple]). For all sufficiently large t , there are λ_1∈B(λ_*,r_0/16) and nonempty compact convex sets K_i^1⊂B(v_i,τ/2) , i=1,2,3 , such that g∈∂P_1^t(λ_1) =(K_1^1∪K_2^1∪K_3^1). All vectors in the K_i^1 …

The function P_*(y)-g·y has its unique minimum at λ_* . In fact P_*(y)-P_*(λ_*)-g·(y-λ_*) =_i(v_i-g)·(y-λ_*)≥γ_0|y-λ_*| for some γ_0>0 , because the triangle of the v_i contains a ball about g . Minimize P_1^t(y)-g·y over B(λ_*,r_0/16) . Lemma 9 shows that all minimizers lie in the interior for large t and converge to λ_* as t→∞ . Choose one and call it λ_1^t . Interior minimality gives g∈∂P_1^t(λ_1^t) . It remains to locate the slopes generating this subdifferential. We claim that any limit of gradients ∇P_1^ t_j (β_j,h_j) , at differentiability points with t_j→∞ and (β_j,h_j)→λ_* , belongs to v_1,v_2,v_3 . Choose a maximizing density d_j at each such point. Apply Lemma 11 ([Supports under a quadratic transform]) with Q=q_ t_j and a=A , choosing e_j∈-∂_β^+H_ t_j (β_j,d_j) . With z_j=h_j+β_jAd_j , it gives ∇P_1^ t_j (β_j,h_j) =(e_j+(Ad_j^2)/(2),d_j), and (e_j,d_j)∈∂q_ t_j (β_j,z_j). The shifted support controls the original first-coordinate slope e_j before the explicit increment Ad_j^2/2 is added. The maximizing densities are bounded by Lemma 9; hence the z_j stay in a compact interval. The locally uniform limit q_* is independent of β , so every cluster of the subgradients in has first coordinate zero. Therefore e_j→0 . The same lemma puts every density cluster in 0,p,4p , and proves the claim. Subgradients of P_1^t near λ_* are bounded uniformly for large t , by. If G_ P_1^t (λ_1^t) were not contained in ⋃_iB(v_i,τ/2) for arbitrarily large t , choose such a sequence of reachable gradients. Approximate each by a gradient at a differentiability point within distance 1/t of λ_1^t , with gradient error tending to zero. A bounded subsequence then contradicts the preceding claim. Thus, for all sufficiently large t , the three intersections G_ P_1^t (λ_1^t)∩B(v_i,τ/2) cover the reachable-gradient set. Let K_i^1 be their convex hulls. They are compact and remain inside their respective balls. Formula gives the equality in. None of the three sets is empty: otherwise g would lie in the convex hull of two of the larger balls U_i . Their density coordinates are nonnegative because P_1^t is nondecreasing in h . Finally, choose one point in each K_i^1 . Their triangle contains B(g,γ) by the fixed geometry of the U_i , and is contained in ∂P_1^t(λ_1^t) .