an inequality from geometry

Hello everybody!

(I write the mathematical formulas in LATEX format)

$n>=2$ is a natural number,

$H$ is the hyperplane consisting of all vectors in
$R^n$ the sum of
the components of which is zero, that is

$$H=\{ x\in R^n s.t. \sum_{i=1}^{n} x_i=0 \}$$

let's denote for ease of writing the cube

$$T=[-1,1]^n$$

I am interested if an inequality like the following holds

$$\forall x\in H$$
$$dist(x,T \cap H)\leq C dist(x,T)$$

for a constant $C$ not depending on $x$.

thanks in advance,
karen

.

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