Maximam of finite sums

Question

It is not so difficult to prove the following estimate: Let $\Gamma \subset \mathbb{C}$ be a finite set.

$$ \sum_{z \in \Gamma} |z| \leq \pi \max_{\Gamma^* \subset \Gamma}\Big|\sum_{z \in \Gamma^*} z\Big|. $$

Can we generalize this inequality to the general Banach spaces? More precisely, let $X$ be a Banach space, and $\Gamma \subset X$ be a finite set. Does the following estimate hold?

$$ \sum_{x \in \Gamma} \|x\|_X \leq C \max_{\Gamma^* \subset \Gamma}\Big\|\sum_{x \in \Gamma^*} x\Big\|_X, $$ where $C$ does not depend on $\Gamma$.

Answer 1

Partial answer: if $X$ is infinite dimension, let $\left\{e_n: n\in \mathbb N\right\}$ infinite orthonormal family of $X$ and let $\Gamma_n = \left\{e_k: k\le n\right\}$ then it is clear that if $C$ exists, $C^2\ge n$.