Is there a common name for the property $f(\lambda x+(1-\lambda)y)\leq f(x)+f(y)$?

Question

For $X$ a vector space and $f:X\to\mathbb{R}$, I'm considering the property

$$f(\lambda x+(1-\lambda)y)\leq f(x)+f(y) \qquad\forall x,y\in X, \;\forall\lambda\in[0,1].$$

So it's like convexity but weaker (actually it forces $f\geq0$ so it's not strictly weaker. Thanks Robert Israel!). I tried googling things like weak convexity, quasi-convexity, subconvexity, et cetera et cetera, but nothing pops up (and to my surprise quite a few of these have actual definitions).

Answer 1

It's not weaker. Note that (taking $\lambda = 0$) you must have $f(x) \ge 0$. Convex functions don't have to be nonnegative.