Question

Question: If A and B are convex sets, prove that \(A + B\) is convex.

Short answer

\(A + B\)is convex.

Step-by-step solution

Make assumptions for x and y \(\left( {{\bf{x}},{\bf{y}} \in A + B} \right)\) 

Let \({\bf{a}}\), \({\bf{c}}\) are in A, and \({\bf{c}}\), \({\bf{d}}\) are in B, such that \({\bf{x}} = {\bf{a}} + {\bf{b}}\), \({\bf{y}} = {\bf{c}} + {\bf{d}}\).

Write w as the combination of x and y

For any \(t\), let \(w = \left( {1 - t} \right){\bf{x}} + t{\bf{y}}\).

So, it can be written as shown below:

\(\begin{array}{c}w = \left( {1 - t} \right){\bf{x}} + t{\bf{y}}\\ = \left( {1 - t} \right)\left( {{\bf{a}} + {\bf{b}}} \right) + t\left( {{\bf{c}} + {\bf{d}}} \right)\\ = \left( {\left( {1 - t} \right){\bf{a}} + t{\bf{c}}} \right) + \left( {\left( {1 - t} \right){\bf{b}} + t{\bf{d}}} \right)\end{array}\)

As A is convex, so \(\left( {1 - t} \right){\bf{a}} + t{\bf{c}} \in A\). Similarly, B is also convex\(\left( {1 - t} \right){\bf{b}} + t{\bf{d}} \in B\). So, \(w\) is in \(A + B\).

So, \(A + B\)is convex.