Uniform convergence of convex functions - references

2012-03-30T04:25:05Z

Inspired by the following question on stackexchange: http://math.stackexchange.com/questions/126142/uniform-convergence-of-convex-sequence-of-functions, I thought of asking whether anyone knows of original (or close to...) references for the following folklore result.

It is known that if $\Omega$ is a compact subset of $\mathbb{R}^n$ which is convex, and if ${f_n}$ is a sequence of convex, continuous functions on $C\subset\Omega$, where $C$ is compact and does not intersect $\partial \Omega$, converging to some continuous $f$ on $\Omega$ pointwise, then the convergence is actually uniform.

Does anyone know of original (or close to...) references for this result? I found a paper (here: http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=ivm&paperid=2536&option_lang=eng) from the '60s (which is in Russian, and here is an abstract in English: http://www.ams.org/mathscinet-getitem?mr=183835) which talks about extensions of this result to include (a subset of) $\partial \Omega$ in $C$, but this paper doesn't mention any references to the original result. By the way, in that paper the author introduces some conditions on $\partial \Omega$ (in particular, he argues that if the curvature of $\partial \Omega$ develops degeneracies, but one can control the order of degeneracy sufficiently well, then the result can be extended, in some sensible way [I didn't spend much time reading into details]).

Uniform convergence of convex functions - references - MathOverflow

Uniform convergence of convex functions - references