TITLE: Surfaces that can be rolled by a ball
QUESTION [7 upvotes]: Let $S$ be a smooth solid body in $\mathbb{R}^3$,
and $B$ a ball of radius $r$.
Say that $B$ is in contact with $S$ if 
(1) they share a point $x$
that is on the surface of each, 
$x \in \partial S$ and $x \in \partial B$,
and (2) neither penetrates the other,
$\operatorname{int} S \cap \operatorname{int} B = \varnothing$.
Say that $S$ can be rolled by $B$ if,
for every pair of points $x,y$ on the surface of $S$, there
is a path $\rho$ connecting $x$ to $y$ on $\partial S$
such that $B$ can be placed in contact at every point $z \in \rho$.
In other words, the ball $B$ can roll between any pair of points of
the surface without ever penetrating $S$.

          


My question is:

Characterize the bodies $S$ that can be rolled by a ball of radius
    $r$.

A necessary condition is that the Gaussian curvature at any point 
on the surface of
$S$ must be $\ge -\frac{1}{r^2}$.
Could that also be sufficient, or are there global obstructions
such that $B$ could be in local contact but penetrate away from
the contact point?
Is the characterization different for surfaces of genus zero than
for surfaces with handles?
Any insights, speculations, or literature pointers would be
appreciated.
Thanks!

Update.  Here is my interpretation of Anton's example:

REPLY [3 votes]: The property you are asking for is called global radius of curvature. The notion combines the condition of lower bound on the principal curvatures as well as the global condition of no interpenetration of a neightborhood. This notion is very well adapted to the calculus of variations.
The heuristic definition from the paper of P. Strzelecki and H. von der Mosel: Global curvature for surfaces and area minimization under a thickness constraint

The main idea can be sketched as follows: Take a continuous parametric surface $X : \mathbb{R}^2\supset \mathbb{B}^2 \to \mathbb{R}^3$ (with possibly infinite area) which possesses a tangent plane on a dense subset $G \subset \mathbb{B}^2$ which may even have zero measure. Consider the radii of all spheres touching the surface  $X(\mathbb{B}^2)$ in one of these points $X(\omega)$,  $\omega \in G$, and containing at least one other point of the surface. We define the infimum of these radii as the global radius of curvature $\Delta[X]$ of the surface $X$ . It turns out that a positive lower bound on $\Delta[X]$ serves as an excluded volume constraint for the surface $X$.