Show that an affine function on a convex and compact set $\Omega \subset \mathbb{R}^d$ is convex?

Question

Please check this out Prove the supremum of the set of affine functions is convex

The answer generalizes without proof that ''every affine function $f_i$ is convex'' on $\Omega \subset \mathbb{R}^d$. How to show that $f_i$ is convex on $\Omega \subset \mathbb{R}^d$.

I know there are $a_i$ $\in$ $\mathbb{R}^d$ and $b_i$ $\in$ $\mathbb{R}$ such that $f_i = a_i.x +b_i$ for all $x \in \Omega$

thanks, so using the definition from wiki http://en.wikipedia.org/wiki/Convex_function#Definition i get 0 $\leq$ 0. Is that right? — sarah, May 31 '13 at 03:56

score 3 · Answer 1 · answered Jul 13 '13 at 04:27

Yes, using whatever definition of convexity on an affine function should give you an inequality of the form $A\le A$, which is trivially true.

By the way, there is no need to assume the domain compact (as in the title of the question): convexity has to do with linear structure, but nothing to do with topology.

Show that an affine function on a convex and compact set $\Omega \subset \mathbb{R}^d$ is convex?

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