The explicit radial collar is smooth into the disk manifold

For every natural number $m$, including $m=0$, let $E=\mathbb R^{m+1}$ with its Euclidean norm, $D=\{x\in E:\|x\|\leq1\}$, and $S^m=\{u\in E:\|u\|=1\}$, with their subspace topologies. The specified charted structure on $D$ has model half-space $H=\{y\in E:y_0\geq0\}$ and atlas consisting of the chart $x\mapsto x+2e_0$ on $\|x\|<1$ and, for each $p\in S^m$, the chart $x\mapsto(1-\|x\|,\sigma_{-p}(x/\|x\|))$ on $x\neq0$ and $x\neq-\|x\|p$, with target $\{y\in H:y_0<1\}$; $\sigma_{-p}$ is stereographic projection from $-p$, followed by the linear isometry to $\mathbb R^m$ chosen in the imported sphere construction. The preferred chart at $a$ is the first chart if $\|a\|<1$, and the boundary chart with $p=a$ if $\|a\|=1$. The function $c:S^m\times[0,1]\to D$ given by $c(u,t)=(1-t/2)u$ is $C^\infty$ in the manifold sense. Its domain uses the product of the standard stereographic sphere structure modeled on $\mathbb R^m$ and the closed interval's manifold-with-boundary structure modeled on $[0,\infty)$, and its codomain uses the stated disk structure modeled on $H$. This asserts smoothness at every domain point, including $t=0$ and $t=1$ and, when $m=0$, both points of $S^0$; the stated regularity is infinite differentiability.