Proof by contradiction. Assume that the line segment between points \(P,Q\) has maximum pairwise distance, and that \(Q\) does not lie on a vertex. Let the point on the boundary of our hull found by extending the line \(PQ\) be denoted \(Q'\). This boundary segment is defined between two vertices on our convex hull which we refer to as \(A\) and \(B\). See \(\mathbf{Fig. 1}\) for clarification.
Clearly \(|PQ'|>|PQ|\). In the following section we show that there is always a line-segment longer than \(|PQ'|\). By the transitive property of inequality this segment must also be longer than \(PQ\), a contradiction.