I was reading the book 'Optimal Statistical Decisions' by DeGroot. I came across the following claim, without proof:
G is a convex set in $R^k$ that is the convex hull of a finite number of points. A point y ∈ G belongs to the Bayes boundary of G if there is no point x ∈ G such that for every i = 1, . . ., k, $x_i$ < $y_i$. Then if y is on the Bayes boundary of G, then there exists a supporting hyperplane ⟨a, x⟩ = c to G at y such that a ≥ 0. (Assuming a is such that for every z ∈ G, ⟨a, z⟩ ≥ c.)
I tried proving it, and while it seems obvious to realise, I am not able to prove it formally. Could someone help me with it?