Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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..

share|cite|improve this question
up vote 2 down vote accepted

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 $(-z, 1)$.

share|cite|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.