I am not a mathematician, but the problem seems easier to intuit. Like a lot of math problems, a formal proof is annoying but I definitely think a regular joe can convince himself why this works.

Start at i,j with heights h_i < h_j. Then we know that if i is involved in the solution, it must be paired with j.

We know this because if it's paired with any other point, the other point is either: a. shorter than h_i, which means it's shorter than h_j. And since it's also closer to i than j is, the area is smaller. b. equal or taller than h_i. But this will never make a larger area since the height will be limited by h_i, but the distance smaller than the distance from i to j.

So we can stop considering i as part of the solution space.

Of course that doesnt make it a good interview question. It relies on a brilliant stroke of insight, rather than relying on established data structures or algorithms.