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?


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