Last semester I attended an introductory course to measure theory, i.e. the basic like $\sigma$-algebras, Lebesgue integration theory, an introduction to $L^p$-spaces and to Fourier analysis. The last part of the course was really dense and to some extent rushed. Giving exercises on several important concept (also sometimes proofs) was not possible. One part which was covered in a few lessons, was the uniform convexity of $L^p$-spaces, mainly Hanner's inequality was proven. I have no idea for what the concept of uniform convexity is good for (I think it is used in functional analysis). One thing is, that every uniformly convex Banach space is a reflexive Banach space (since $L^p$-spaces are Banach spaces for $1 \leq p \leq \infty$). So my question is:
- Why is it good to be a uniformly convex space? Especially in this particular case of $L^p$-spaces.
- Are there any other applications of Hanner's inequality?