I am interested in the following. Given are two Banach spaces over $\mathbb{C}^n$ with respective norms $\|\cdot\|_0$ and $\|\cdot\|_1$.
Suppose it holds that
$\tfrac{1}{2}\left(\|x+y\|_0^p + \|x-y\|_0^p\right) \leq \|x\|_0^p + C\|y\|_0^p$
and
$\tfrac{1}{2}\left(\|x+y\|_1 + \|x-y\|_1\right) \leq \|x\|_1 + C\|y\|_1$
Now, I would like to prove that for $\theta \in (0,1)$, we have that
$\tfrac{1}{2}\left(\|x+y\|_\theta^{p_\theta} + \|x-y\|_\theta^{p_\theta}\right) \leq \|x\|_\theta^{p_\theta} + C\|y\|_\theta^{p_\theta}$.
where $\|\cdot\|_\theta$ is the norm obtained by (complex) interpolation between $\|\cdot\|_0$ and $\|\cdot\|_1$. (and $\tfrac{1}{p_\theta} = \tfrac{1-\theta}{p} + \tfrac{\theta}{1}$)
I do believe this inequality ought to follow from Riesz-Thorin. But I do not understand how this inequality follows from the definitions of complex interpolation in this simple setting. In particular, I only understand Riesz-Thorin in the setting of $L_p$ functions since the norms can be represented nicely in terms of products with their duals. Here, the norm $\|\cdot\|_\theta$ is given in terms of $\|\cdot\|_0$ and $\|\cdot\|_1$ which makes it difficult to compare the two expressions.
Any help is much appreciated, I will put a bounty on this once I can.