# convex lsc function affine minorant theorem proof

How to show the following:

f:R^n->R

A lower semicontinuous convex function f equals the pointwise supremum of all its affine minorants.

Thank you!

-
Also here you should specify the domain of definition of $f$. Moreover, the answer lies the separating the epigraph of $f$ from any point not in the epigraph by a hyperplane (using, e.g. a variant of Hahn-Banach). –  Dirk Feb 11 '13 at 10:41
Could you give more details how do you use affineness lsc and convexity there? –  Salih Ucan Feb 12 '13 at 4:32
The epigraph is closed iff the function is lsc. Affine functions from $\mathbb{R}^n\to\mathbb{R}$ correspond to hyperplanes in $\mathbb{R}^{n+1}$. Convexity is needed in the separation theorems. –  Dirk Feb 12 '13 at 7:19