At one of my lectures recently (https://github.com/joanbruna/MathsDL-spring18/blob/master/lectures/lecture13.pdf , on the slide titled Parametric vs Manifold Optimization), I came across the definition of ball-connected set:
A set is ball-connected if its intersection with balls of any radius and is a single connected component.
I remember it being said that this definition is weaker than convexity. What I don't understand is that how is this any different from convexity? Can't we just make the radius $r$ large enough that the intersection with this set is just a hyperplane cutting the set, and if its still connected for every such hyperplane then it is convex, right?
If you consider unit ball with its centre removed then it's not convex.
But it's ball-connected as it's intersection with any radius ball is single connected set.
For example, consider open unit disk in $\mathbb R^2$.
Hence ball-connected is weaker notion than convexity.