Convexity/Examples of convex sets

Example 1

In a two-dimensional vector space, a parallelogram is a set such that in some suitably chosen basis x, y of the space, the set consists of the points ax + by with 0 < a < 1, 0 < b < 1.

All parallelograms are convex. For, given any two points A, B in the parallelogram, we have

A = ax + by

B = cx + dy

with all coefficients being between 0 and 1. An arbitrary point on the line AB is

C = (λa+(1-λ)c)x + (λb+(1-λ)d)y

with 0 < λ < 1. These coefficients are also between 0 and 1, so C is also in the parallelogram.

Example 2

In Euclidean space, a ball, centre O radius r is the set of points within distance r of O, i.e. it is the interior of a sphere or hypersphere. (In two dimensions, a ball is often called a disc.)

All balls are convex. For, given any two points A, B in the ball, we have their distances from O less than r. For an arbitrary point C on AB, C = λA+(1-λ)B so

dist(O,C) < λdist(O,A) + (1-λ)dist(O,B) < r.

Hence C is also in the ball.