Epigraph supported at some point meaning of the sentence

Question

Can you tell me what does the following sentence mean?

Let $z \in \mathbb{R}^d$ and $(-z,1)$ supports epigraph of $f$ at $(x_0,f(x_0))$

Thank you..

Answer 1

Formally it means that $$ f(y) \geq f(x_0) + z' (y - x_0) \ \ \ \ \ \forall y \in domf. $$

If you are looking for a geometrical intuition, you can see that the inequality can be written also as $$ \left(y - x_0, f(y) - f(x_0) \right) \left(\begin{array}{c} -z \\ 1 \end{array} \right) \geq 0 $$ which tells you that the epigraph of $f$ is contained in halfspace defined by the hyperplane that passes by $(x_0,f(x_0))$ and is normal to vector $(-z, 1)$.