Checking Convexity of a Set - Nonadditivity & Nondivisibility

[Kinetica]

A set is convex if it's additive and divisible. To find out the convexity you just pick any two points and draw a tangent line. If the line lies within the area - voila, the set is convex. Similarly, if the set is nonconvex, you just show that a part of any tangent line can lay outside of the set.

But in the case of nonconvexity, how do I show that it is also nonadditive and nondivisible?

I've got a set that looks like a flat donut where the donut hole is not a part of the set. I already showed that the set is nonconvex by drawing a tangent line. I do struggle to show (without complex proofs), that the set is also nondivisible and nonadditive.

Any help is greatly appreciated.

 

[lippyka]

To show that the set is nonadditive, you can try to pick two arbitrary points from the set and show that their sum does not belong to the set. To show that the set is nondivisible, you can try to pick an arbitrary point from the set and show that its halves do not belong to the set.