Is is true that, in any convex quadrilateral $\mathcal{Q}$, the longest distance between any two points of $\mathcal{Q}$ is attained when we join two non-adjacent vertices of $\mathcal{Q}$?
I suppose this fact (or its negation) is well-known, but I can't find it in the bibliography I have at hand.
Thank you very much for your help.
No, that isn't true.
E.g. take the quadrilateral with vertices $(0,0), (1,1), (1,3), (0,4)$.
The longest distance is between $(0,0)$ and $(0,4)$, i.e. two adjacent vertices.
What is true is that the longest distance is between two vertices.
EDIT: I first added that probably these two vertices had acute angles, but this is false.