In Hilbert space modeled by $L^2([0,1])$ we can define a set $B^+=\{x\in B(0,1): x(t)\geq0 \quad \forall t\in [0,1] \}$ and $S^+=\{x\in S(0,1): x(t)\geq0 \quad \forall t\in [0,1] \}$ where where $B(0,1)$ and $S(0,1)$ are, respectively, a closed unit ball and an unit sphere centered at origin. $S^+$ is a Lipschitz retract of $B^+$. It can be shown that a Lipschitz constant of this retraction cannot be smaller than $\frac{\pi}{2}$. The best upper bound I know is $2\sqrt{2}$.
$T(x(t))=1-\|x\|+x(t)$ is 2-Lipschitz and $\|T(x(t))\| \geq \frac{1}{\sqrt{2}}$, so $\frac{T(x)}{\|T(x)\|}$ is a $2\sqrt{2}$ Lipschitz.
Is there any known better bound on retraction constant of $B^+$ into $S^+$?
