Criterion for determining when uniform convergence of multivariate convex functions is convex

Question

Suppose we have a sequence of convex multivariate functions

$f_a(\vec{v}): S \to \Bbb R$

defined on some open convex subset $S$ of $\Bbb R^n$, which converge uniformly to some limit function $f_\infty(\vec{v})$.

Question: What criteria exist to establish that $f_\infty(\vec{v})$ is also convex?

I think this question shows this need not be true in general, but is there some simple criterion for when this is true?

FYI, while I have searched and found a lot of questions about uniform convergence and convex functions, I haven't found a question addressing such criteria in the multivariate case. Apologies if this has been asked already.

Answer 1

Pointwise limits of convex functions are convex (by definition) and hence uniform limits too. The post you have referred to is irrelevant to this question.

If $S$ is a convex subset of a vector space $V$ over the reals then a function $f:S \to \mathbb R$ is called convex if $f(tx+(1-t)y) \leq tf(x)+(1-t)f(y)$ whenever $x,y \in S$ and $0 \leq t \leq 1$. If $f_n(tx+(1-t)y) \leq tf_n(x)+(1-t)f_n(y)$ and $f_n \to f$ pointwise to $f$ we can simply let $n \to \infty$ to get $f(tx+(1-t)y) \leq tf(x)+(1-t)f(y)$.