Prove that the Minkowski functional associated with a set $K$ satisfies the triangle inequality if and only if $K$ is convex.

Question

Taking analysis, have a problem set that's tilting toward topology. The problem asks for proof that the Minkowski functional associated with a set $K$ satisfies the triangle inequality if and only if $K$ is convex.

Really just don't know where to start. Any help would be appreciated.

Answer 1

Hint: It seems the proof of the equivalence can be straightforward, so you may start from the formulation of the statements which you need to prove and remembering the definition of Minkowski functional.