Let $D$ be some connected convex $C^2$ domain in $\mathbb{R^n}$ and $y\in \overline{D}^c$. Let $d$ be the distance to the boundary function of $D$. Assume $d(D,y)=|y-x_y|$ for all such $y$ for a unique $x\in \partial D$.
First question: is there a name for this condition or a known condition that guarantees it?
For a given $y$ in the exterior of $D$, define the line $I:x_y+ r y\quad, r\in \mathbb R$.
If we denote by $n$ the unit normal, is it true that $n(x_y) \subset I$? My intuition says yes, as the distance minimizer should be realized by the orthogonal projection but I am not sure how to conclude. I know that $n(x_y)=-\nabla d(x_y)$.
Since $x_y$ is closest to $y$ in $\delta D$, then the closed ball $B$ centered at $y$ and of radius $|y-x_y|$ intersects $\overline{D}$ only at $x_y$.
That means that the tangent plane $\pi$ to $B$ at $x_y$ is also tangent to $D$ at $x_y$. This means that that normal vector $n(x_y)$ is orthogonal to $\pi$, but so is vector $\vec{yx_y}$. So $n(x_y)$ is parallel to the line $I$.