Exercise 5 in chapter 2 of Lieb and Loss's Analysis asks the reader to show that Hanner's inequality,
$$\|f+g\|_p^p + \|f-g\|_p^p \ge (\| f\|_p + \|g \|_p )^p + |\| f\|_p - \|g \|_p |^p,$$
which is valid in $L^p(\mathbb R^n)$ for $1\le p \le 2$, "shows that the unit sphere is 'uniformly smooth', i.e., it has no corners."
I wasn't sure what this statement meant, so I looked up the definition in the Banach space literature and found a few equivalent ones, along with proofs of much more general statements that implied this one. However, I can't figure out what Lieb and Loss are getting at. None of the proofs I found seem appropriate for a textbook exercise, because they go through several intermediate characterizations of uniform smoothness.
Is there a way to interpret this claim so that it has a relatively immediate proof from Hanner's inequality?