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There is a systematic approach to this typo of problems which finds the best possible constants (when $n\to \infty$) but requires plenty (but easy) computations. I will mention the steps and if somebody is interested I can provide more details What are we looking for?  \sup_{x,y \in L^{p}}\{ \|x+y\|_{L^{p}} : \|x\| _{L^p}=1, \|y\| _{L^p}=1, \| x-y\| ...
Here is a simple short proof of one of the first main questions. Claim. $C_p < 2$. Proof. Let $C_p$ be above. We look at two cases: (i) $1 < p < 2$, and (ii) $p > 2$. Case (i): From this short note we know that \begin{equation*} \|x+y\|_p^2 \le 2(\|x\|_p^2 + \|y\|_p^2) + (1-p)\|x-y\|_p^2. \end{equation*} Using the hypothesis, ...