Why, in functional analysis, is so important to calculate best constant in an embedding inequality?


Cross-posted from "<https://math.stackexchange.com/questions/727690/understanding-reasons-for-best-constant-in-inequalities>".



Understanding reasons for best constants in inequalities
A simple reason, among many others: for instance, you would like to know whether a certain functional is bounded below, because you are looking for a minimizer of it, and you can prove some inequality of the form $f(u)\ge \|u\|\_1^\alpha-c\|u\|\_2^\alpha +b $.


