Let $p_1,\dotsc,p_n$ be positive integers. Define the complex ellipsoid $$\Omega(p_1,\dotsc,p_n)=\left\{(z_1,\dotsc,z_n)\in\Bbb C^n:\sum\limits_{i=1}^n{\left|z_i\right|^{2p_i}}<1\right\}.$$ I want to prove that the set $\Omega(p_1,\dotsc,p_n)$ is convex.
I have already proved that $\Omega(p_1,\dotsc,p_n)$ is convex for $p_1=p_2=\dotsb=p_n$ using the Minkwoski's inequality. But I don't find any way to prove the convexity for distinct $p_i$'s. Help is appreciated.
First we have $$ \lvert (1-t)z+tw \rvert^{2p} \leqslant (1-t)\lvert z \rvert^{2p} + t \lvert w \rvert^{2p}, \tag{1} $$ for any $p>1/2$, either by your previous result with equal $p_i$, or by the usual convexity of $z \mapsto \lvert z \rvert^{2p} $. But then, supposing $z,w \in \Omega$, we have $$\begin{align*} \sum_{i=1}^n \lvert (1-t)z_i+tw_i \rvert^{2p_i} &< \sum_{i=1}^n \left( (1-t)\lvert z_i \rvert^{2p_i} + t\lvert w_i \rvert^{2p_i} \right) \\ &= (1-t)\sum_{i=1}^n \lvert z_i \rvert^{2p_i} + t \sum_{i=1}^n \lvert w_i \rvert^{2p_i} \\ &< (1-t)+t=1, \end{align*} $$ by using (1), linearity of the summation, and $z,w \in \Omega$.