Dear Sudhanshu Aggarwal ,
In Euclidean spaces:
Every convex set is connected, but the converse is not true.
Example: The ring region K = {(x,y), 1 < x2 + y2 < 9} in the plane is connected but not convex.
Proof: The points A(2, 0) and B(-2, 0) belong to K, but the line segment AB doesn't belong to K, hence non-convex region.
K is pathwise connected and connected since for any points A and B
in K can be joined with a path.
Best wishes
Dear Sudhanshu Aggarwal ,
In Euclidean spaces:
Every convex set is connected, but the converse is not true.
Example: The ring region K = {(x,y), 1 < x2 + y2 < 9} in the plane is connected but not convex.
Proof: The points A(2, 0) and B(-2, 0) belong to K, but the line segment AB doesn't belong to K, hence non-convex region.
K is pathwise connected and connected since for any points A and B
in K can be joined with a path.
Best wishes
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