Lipschitz Constant of Standard Projection

Question

Let $D> d\geq 1$ be positive integers, let $X\triangleq \{x \in \mathbb{R}^D: x_{i}=0\, \forall d<i\leq D\}$, and define the canonical projection $$ \pi^D_d(x)\triangleq (x_i)_{i=1}^D\mapsto (x_1,\dots,x_d,\underbrace{0,\dots,0}_{D-d-times}), $$ be the canonical projection of $\mathbb{R}^D$ onto $X$; clearly this is Lipschitz with constant at-least $1$. Is there a sharper constant, since in the dummy example I look at it seems to be a strict contraction...

Shouldn't the Lipschitz constant depend on $D$ and on $d$?.

Also, nice to ask my first stack-exchange question.

Answer 1

If the domain of the projection is all of $\mathbb{R}^D$, then on the subspace say $S$ that it projects onto, the projection is just the identity, which obviously has Lipschitz constant equal to 1.

Answer 2

$1$ is the smallest constant since $$\|\pi^D_d(1,0,\ldots, 0) - \pi^D_d(0,0,\ldots, 0)\| = \|(1,0,\ldots, 0) - (0,\ldots, 0)\|$$