Unicity of the projection on a closed convex subset of $L^p$

Question

The original problem (not full) statement is the following :

Let $(\Omega, \mathfrak{B},\mu)$ a measured space and $p>1$ and let $C$ be a closed convex subset of $L^p$.

For $u\in L^p$ let $d_u:=\inf\{\|\phi-u\ \|, \phi \in C\}$

Through some questions, the problem shows the existence of a $\phi \in C$ such that $\|\phi-u\ \|=d_u$.

The problem I have is with unicity. I read somewhere on MSE that the $L^p$ norm is strictly convex, which implies unicity.

As far as I know, no norm is strictly convex, so the previous affirmation might be wrong so how could we deal with unicity.

Answer 1

Hint:

For $p\in(1,\infty)$, assume there are two different solutions $\phi,\psi\in C$. Then, by convexity of the norm, one can show that $(\phi+\psi)/2$ is closer to $u$ than $\phi$ or $\psi$.

In order to show the last part, you have to use the fact that for real numbers $a,b$ it holds that $$ | (a+b)/2 | ^p = (|a|^p+|b|^p)/2 \quad\Leftrightarrow\quad a = b. $$