Simultaneous version of Triangle inequality
Theorem 1. If $\alpha_1,\dots,\alpha_m$ are arbitrary complex numbers, $m\geq2$, and if $H\in\mathbb{N}$, then there exist numbers \begin{align} a_k\in\mathbb{Z}, \quad \lvert a_k\rvert\leq H, \quad k=1,\dots,m, \quad \max_{1\leq k\leq m} \lvert a_k\rvert>0 \end{align} such that the linear form \begin{align} L=a_1\alpha_1+\cdots+a_m\alpha_m \end{align} satisfies the inequality \begin{align} \lvert L\rvert\leq cH^{1-\tau m} \end{align} where $\tau=1$ if all of the $\alpha_k$ are real and $\tau=\frac{1}{2}$ if one or more of them is complex, and $$c=\left\{ \begin{array}{ll} \sum_{k=1}^m \lvert \alpha_k\rvert, & \quad \text{if}\;\tau=1 \\ \sqrt{2}\sum_{k=1}^m \lvert \alpha_k\rvert, & \quad \text{if}\;\tau=\frac{1}{2} \end{array} \right.$$ Proof of Theorem 1. The theorem is obvious if $H=1$ by the generalization of Triangle inequality; so we suppose that $H\geq 2$. We consider all possible linear forms $L$, where the $a_k$ independently run through all int...