The author addressed this point in a comment on the article:
> True, it’s not a rigorous proof. You can find a rigorous proof here:
> “A Simple Noncalculus Proof That the Median Minimizes the Sum of the Absolute Deviations.” Neil C. Schwertman, A. J. Gilks, and J. Cameron. The American Statistician, Vol. 44, No. 1 (Feb., 1990), pp. 38-39.
> True, it’s not a rigorous proof. You can find a rigorous proof here:
> “A Simple Noncalculus Proof That the Median Minimizes the Sum of the Absolute Deviations.” Neil C. Schwertman, A. J. Gilks, and J. Cameron. The American Statistician, Vol. 44, No. 1 (Feb., 1990), pp. 38-39.
That paper fits on one page, not counting references. Here's a link: https://tommasorigon.github.io/StatI/approfondimenti/Schwert...