Few questions about a reconstruction theorem. I'm reading through the proof of the following theorem
Lemma 1.1. (Feature Ball). If a $d$-ball $B = B_{c,r}$ intersects a $k$-manifold $\Sigma \subset \mathbb{R}^d$ at more than one point where either $(i)$ $B \cap \Sigma$ is not a $k$-ball or $(ii)$ $bd ( B \cap \Sigma)$ is not a $k-1$-sphere, then a medial axis point is in $B$.
Proof. First we show that if $B$ intersect $\Sigma$ at more than one point and $B$ is tangent to $\Sigma$ at some point, $B$ contains a medial axis point.
Question 1. Why is studying such a case necessary in the first place? since it's not even mentioned in the theorem statement, unless this happens with case (i) or (ii)?
Let $x$ be the point of this tangency. Shrink $B$ further keeping it tangent to $\Sigma$ at $x$. We stop when $B$ meets $\Sigma$ only tangentially.
Question 2. I can picture exactly what's happening in 2D or even 3D, and I can imagine what happens with dimension $k > 3$, however I can only picture this I'm missing exactly what transformation is defined to perform this shrinking.
They're saying to show that $B$ contains a medial axis point, it suffices to show that $B$ is tangent to $\Sigma$ at some point.
Let $c$ be the center of $B$ and $x$ the point of tangency. Let $y_t=(1-t)x+tc$, where $t\in[0,1]$, be a point on the line segment between $x$ and $c$. We want to pick $t$ such that $\varphi(t) = d(y_t, \Sigma\setminus\{x\}) - \|y_t - x\| = 0.$ But $\varphi$ is continuous with $$\varphi(0) = d(x, \Sigma\setminus\{x\}) - \|x-x\| > 0$$ $$\varphi(1) = d(c, \Sigma\setminus\{x\}) - \|c-x\| < 0.$$