I'd notice that this must also describe the best shape of a glass to keep a beverage hot. But, unlike shot glasses, tea and coffee cups look very different.
As you correctly point out the problem is entirely symmetrical to that of keeping a hot beverage as hot as possible. If we take the different problem of trying to cool a hot beverage, the optimum, pathological solution would be a glass with 0 height and infinite width --studying this problem with the additional constraint that width be limited to some predefined value might make for an interesting followup article.
I would add the constraint that the surface is convex, too. Without that, one could build some 3D fractal, and get infinite area for any given volume.
Alternatively, complicate things by taking the width of the glass into account or model convection more realistically (a square meter of glass close to other glass of similar temperature will not lose much heat)
I guess that the ratio h_open/h_closed is much bigger for hot beverages, due to evaporative cooling. Of course, this would mean that the optimum shape is even thinner than those that were calculated.
But maybe the shape of tee-cups and coffee mugs is optimal, in its own way? I certainly appreciate that beverages that need quite high temperatures for brewing are cooled down quickly to a drinkable temperature.
