Zoltán Daróczy, Zsolt Páles: Convexity with given infinite weight sequences.
Stochastica: revista de matemática pura y aplicada, ISSN 0210-7821, Vol. 11, No. 1, 1987, 5-12, link to pdf.
Lemma 1. Let $D\subseteq\mathbb R^n$ be a convex compact set. Let $f:D\to\mathbb R$ be an $\alpha$-convex function for some fixed $\alpha\in(0,1)$, i.e. assume that
$$f(\alpha x+(1-\alpha)y)\le \alpha f(x)+(1-\alpha)f(y) \qquad (\ast)$$
holds for all $x,y\in D$. Then $f$ is also Jensen convex, i.e. $(\ast)$ is satisfied with $\alpha=\frac12$.
According to the autors, the same result was obtained earlier by N. Kuhn in the paper A note on t-convex functions, but I was not able to find this paper online.
Based on the above result, it suffices to consider $t=\frac12$. It was explained in this answer
that some form of AC is needed to construct non-convex Jensen convex function. (Since every measurable Jensen convex function is convex.)