For each of the following sets, give the coordinates of two points P and Q which are in the set, but the line from P to Q goes outside the set.

Question

In this question, we show that certain sets are not convex.

For each of the following sets, give the coordinates of two points P and Q which are in the set, but the line from P to Q goes outside the set.

{(x,y)∣x^2+y^2≥4,(x−2)^2+y^2≤4}

Answer 1

We have two radius 2 circles, one centered at (0, 0) and the other centered at (2, 0). The concave set is the shaded area inside the second circle but outside the first circle.

P = (–3/2, 3/2) and Q = (3/2, 3/2), for example, are a pair of points in the set, but the line from P to Q goes outside the set.