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