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".
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Sign up to join this communityWhy, 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".
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 $.
In some cases the precise value of the best constant may not be very important in itself, but finding a characterisation of the extremal case (if it exists) or how to approach the extremal value of the constant may give important insight.
If we think about functional inequalities on manifolds, finding optimal constants is sometimes understanding the geometry of the manifold and characterize it. Let me take two basic examples.
1) The Poincare inequality (or Lichnerowicz estimate) on a Riemannian manifold with Ricci curvature bounded from below by $ \rho >0$ writes
$\int_{\mathbb{M}} f^2(x) dx \le \left( \int_{\mathbb{M}} f(x) dx \right)^2 +\frac{n-1}{n\rho} \int_{\mathbb{M}} \| \nabla f \|^2 (x)dx.$
The constant $ \frac{n\rho}{n-1}$ is sharp, because on the $n$-dimensional sphere, the first non trivial eigenfunction achieves the equality. It is remarkable, that it is a way to characterize the sphere. That is, if there exists a function $f$ that achieves the equality, then the manifold needs to be isometric to the sphere (This is Obata theorem)
2) The sharp (tight) Sobolev inequality that says that on on a Riemannian manifold with Ricci curvature bounded from below by $ \rho >0$, for $ p = \frac{2n}{n-2}$
$ \frac{n \rho}{(n-1)(p-2)} \left( \left( \int_{\mathbb{M}} | f |^p(x) dx\right)^{2/p}-\int_\mathbb{M} f^2(x) dx \right) \le \int_\mathbb{M} \| \nabla f\|^2(x) dx.$
attracted quite some attention because it implies the Bonnet-Myers theorem with a sharp constant on the diameter.
There are many other examples of functional inequalities, where optimal constants characterize the manifold. You will for instance find quite a lot of them in Chapter 3 of the following survey by Ledoux http://www.math.univ-toulouse.fr/~ledoux/Zurich.pdf
Because the goal of mathematics is to discover the truth.