Let $X$ be a $d$-dimensional Banach space with norm $\| \cdot \|$ and bases $e _1 , e _2 , \ldots , e _ d$. By the equivalence of all norms in finite dimensional space, there exists $c> 0$ such that $$ \left \| \sum ^{d}_{i=1} \lambda _i e _i \right \| \geq c \sqrt{\sum ^{d}_{i=1} \lambda ^2 _i} $$ holds for any real numbers $\lambda _1 , \lambda _2 ,\ldots ,\lambda _d$. As far as I see the proof, constant $c> 0$ possibly depends on the choice of basis and it is difficult to deduce explicit formula of $c $.
However, in J. Lindenstrauss, Bull. Amer. Math. Soc. 72 (1966), 967–970, the following fact is used: there exist a basis $e ' _1 , e '_2 , \ldots , e '_ d$ such that $\| e '_i \| =1 $ and $$ \left \| \sum ^{d}_{i=1} \lambda _i e' _i \right \| \geq \frac{ \sqrt{\sum ^{d}_{i=1} \lambda ^2 _i} }{d^2} $$ holds for any $\lambda _1 , \lambda _2 ,\ldots ,\lambda _d$. This mean that we can choose a suitable normal basis so that we can take $c = 1 / d ^2 $ above.
Do you know how to prove it or construct such a basis that $c$ only depends on the dimension $d$ (not necessarily $c =1 / d^ 2 $)?
The key is the following fact given in ``A. E. Taylor, A geometric theorem and its application to biorthogonal systems, Bull. Amer. Math. Soc. 53 (1947), 614--616''; let $X$ be a Banach space with norm $\| \cdot \|$ and $Y$ be a $d$-dimensional subspace of $X$. Then there exist $x _ 1 , x _ 2 , \ldots , x _d \in Y $ and $f _ 1 , f _ 2 , \ldots , f _d \in X^{\ast} $ (dual of $X$) such that $\| x _ i \| _X = \| f _ i \| _{X ^{\ast }}= 1$ for all $i = 1, 2, \ldots ,d$ and $$ f _i (x _j ) = \begin{cases} 1~~(i=j),\\ 0~~(i\neq j), \end{cases} $$ (such $x _ 1 , x _ 2 , \ldots , x _d \in Y $ and $f _ 1 , f _ 2 , \ldots , f _d \in X^{\ast} $ is called biorthogonal system).
By using a biorthogonal system, we define planes $P_i := \{ y \in Y ; f _ i (y ) = 1 \}$ for $i=1 , 2, \ldots ,d$. Obviously, $$ P_ i =\left \{ y = x _ i + \sum _{j \neq i} \alpha _j x _ j;~~\alpha _j \in \mathbb{R} \right \} . $$ Then we have $y \in P_i ~ \Rightarrow ~ \| y \| _X \geq 1$ since $\| y \| _X < 1 ~ \Rightarrow ~ | f _ i(y) |<1 $ holds by $\| f _ i \| _{X ^{\ast }} =1$. This implies that $$ \left \| x _ i + \sum _{j \neq i} \alpha _j x _ j \right \| _X \geq 1 $$ holds for every real numbers $\alpha _1 ,\alpha _2 , \ldots , \alpha _n$.
We can see that $x _ 1 , x _ 2 , \ldots , x _d $ is a system of $Y$ satisfying the required property with coefficient $c =1 /d $. Indeed, let $ \lambda _1 , \lambda _2 , \ldots , \lambda _d \in \mathbb{R}$ and $ |\lambda _j | = \max _i | \lambda _i | > 0 $. Then $$ \left \| \sum ^d _{ i =1} \lambda _i x _ i \right \| _X = |\lambda _ j | \left \| x _ j + \sum _{i \neq j} \frac{ \lambda _i }{ \lambda _ j } x _ i \right \| _X \geq |\lambda _ j | \geq \frac{\sum ^d _{i=1 } |\lambda _ i | }{d} \geq \frac{\left ( \sum ^d _{i=1 } |\lambda _ i | ^2 \right ) ^{1/2}}{d} . $$