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I'm working with a problem the epigraph of a real-valued function $f$ is a halfspace $\iff$ $f$ is a real-valued affine fuction.

First, I quickly recall some definitions:

  • A (closed) halfspace is a set of the form $\{x:a^T\textbf{x} \le b \}$ for some $a \in \mathbb R^n,a \ne 0, b \in \mathbb R.$

  • epi$(f)=\{(\textbf{x},t):f(\textbf{x}) \le t \}$.

  • A real-valued function $f$ is affine if $f(\textbf{x})=c^T\textbf{x}+d$ for some $c \in \mathbb R^n,d \in \mathbb R.$

Second, I show my little attempt:

$"\Longleftarrow"$ Assume $f$ is a real-valued affine fuction then

$(\textbf{x},t) \in \text{epi}(f) \Leftrightarrow f(\textbf{x}) \le t$ $\Leftrightarrow c^T\textbf{x}+d \le t \Leftrightarrow c^T\textbf{x}\le t-d$.

The last inequality shows $c^T\textbf{x}\le t-d$, that means $\text{epi}(f)$ is a halfspace. Is that true?

$"\Longrightarrow"$ Assume epigraph of $f$ is a halfspace. We need to prove $f$ is affine.

I have some ideas but they didn't work. How can I continue?

user
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  • The backward direction looks good. For the forward direction, I think you can say that since $epi(f)$ is a halfspace, then $f(x)= a^t x$, which is affine. – IAmNoOne May 07 '14 at 06:02

1 Answers1

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I've attempted to solve the second half of your proof by considering that when $epi(f)$ is a halfspace:

$(x, t) \in$ epi($f) \iff (b, c^T)*(x, t)^T \leq t \iff c^T*x + bt \leq t \iff f(x) \leq t$

which shows that $f$ is equivalent to an affine function.

jaydez
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