Exact coefficient in equivalence of norm in finite dimensional space. 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 $)?
 A: 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
$$
\left \| \sum ^d _{ i =1} \lambda  _i x _ i \right  \| _X
= 
|\lambda _ j | \left \| x _ j + \sum _{i \neq j} \frac{ \lambda _i  }{  \lambda _ j }  x _ i  \right  \| _X
\geq |\lambda _ j | \geq \frac{\sum ^d _{i=1 } |\lambda _ i | }{d}  \geq 
 \frac{\left ( \sum ^d _{i=1 } |\lambda _ i | ^2 \right ) ^{1/2}}{d}  .
$$
