Let $S$ be a $C^2$-regular hypersurface with $S=\partial V$ for some open set $V \subset R^{N+1}$, and let $\nu(P)$ be the exterior unit normal of $S$ with respect to $V$.

Assume that $S$ satisfies the R-sphere condition, that is for every $P\in S$ the tangent balls $B^\pm(P,R):= \{Q \in R^{N+1}: \ |P\pm R \nu(P) - Q|<R\}$ do not contain any points of $S$.

Let the origin $O\in S$ and assume that $\nu(O)=e_{N+1}$. By the implicit function theorem we know that $S$ can be written locally as a graph of function. In particular, there exists $r>0$ and a real valued function $u$ such that $$ \{ (x,u(x)): |x|<r \} \subset S .$$

**Can we conclude that $r=R$ **?

I proved that this is true for $N=1$ as sketched in the following. By writing the touching balls at $O$ we obtain that $$ |u(x)| \leq \rho - \sqrt{\rho^2-x^2},\quad |x|<\rho. $$ If we assume that $\nu((x_0,u(x_0))) \cdot e_2 = 0$ for some $|x_0|<\rho$, then the centers $C^\pm$ of the two touching balls at $(x_0,u(x_0))$ must satisfy $|C^\pm|\geq \rho$ (otherwise one of the two touching balls will contain the origin). By writing down the inequality and using the above estimate on $u$ we find a contradiction.

Unfortunately, it seems to me that the same argument does not work for $N\geq 2$.