Perturbation of tangent ball

Question

As picture below, $A$ and $B$ are two balls, $\partial A\bigcap \partial B=\{k\}$, and $B$ contains $A$. How to show that $$ \forall h\in \partial B,\exists ~\varepsilon > 0 ~st~ A\subset B+\varepsilon\overrightarrow{hk} $$

Answer 1

We shall prove $$A+\epsilon\,\vec{kh}\subset B$$ for a suitable $\epsilon>0$, which amounts to the same thing. Let $a< b$ be the radii of the two spheres. I claim that $$\epsilon:={b-a\over b}$$ does the job.

Proof. Stretch $A$ with stretching center $k\in\partial B$ by the factor of ${b\over a}$. The resulting sphere $A'$ will then coincide with $B$. Now shrink $A'$ with shrinking center $h\in\partial B$ by the factor ${a\over b}$. The resulting sphere $A''$ has radius $a$, and touches $B$ at $h$ from the inside. It is easily checked that the center of $A''$ is given by $c+{b-a\over b}\vec{kh}$, where $c$ denotes the center of $A$.