convex set

A convex set S is a set of points P₁, ... , P_n such that for any two points x and y within S, all the points along the line segment between x and y are completely contained within S.

        _______
      /         \    These boundaries enclose a convex set.  This is because for
     /   x---y   \   any points x and y within the set, the line between them is 
     \           /   completely included in the set.
      \ _______ /

    ____          ____
   |    |        |    |   These boundaries define a non-convex set.  Although
   | x--|--------|--y |   the points x and y are inside the set, the line between
   \    |________|    /   them is not completely contained within the set.
    \________________/