# Why support function describes the (signed) distances of supporting hyperplanes of A from the origin?

The support function of a set $$A \in \mathbb{R}^n$$ is defined as the following

$$S_A(x)=\sup_{y \in A} x^Ty$$ where $$x \in \mathbb{R}^n$$.

In Wikipedia: Support function it says support function describes the (signed) distances of supporting hyperplanes of A from the origin.

What is the intuition behind this, and why it is true?

“Support function of a set” and supremum question talks about this point but does not discuss the reason.

Could you provide with me a picture to understand what is going on?