Intuitive feeling for Convex set

Question

Can anyone please give me some insight about convex set I am aware of the definition of convex set - A set $E$,which is a subset of $R^k$ if $¶.x + (1-¶).y$ belongs to E whenever x,y belongs to E and $0<¶<1$. I am still in the beginning chapters of Rudin,so my knowledge is very basic.

Answer 1

Intuitively a set is convex if every segment you draw between two points in the set is still in the set. For example a disk is convex because if you take two points in the disk then the segment that link these two points is still in the disk.

Answer 2

You can't hide in a convex set. No matter where you are, and no matter where your adversary is, it can see you and you can see it.

In a nonconvex set, on the other hand, it is possible to hide: there are some positions where you and your adversary cannot see each other, because some corner or other obstruction is blocking your view of each other.