The lack of strict convexity and the validity of the Comparison Principle for a simple class of minimizers

Let w∗(x)=a⋅x+b be an affine function in RN, Ω⊂RN, L:RN→R be convex and w be a local minimizer of I(v)=∫ΩL(∇v(x))dx in W1,1(Ω,R) with w(x)≤w∗(x) on ∂Ω in the trace sense. Then w∗ satisfies the Comparison Principle from above, i.e. w(x)≤w∗(x) a.e. on Ω if and only if (a,L(a)) does not belong to the relative interior of a N-dimensional face of the epigraph of L. As a consequence, if F is the projection of a bounded face of the epigraph of L, the local minimizer w∗(x)=max{ξ⋅(x−x0):ξ∈F} satisfies the Comparison Principle from above if and only if dimF≤N−1 or x0∉Ω.