We know the upper contour set of a convex function is not necessarily convex.
However, suppose $f\left(y\right)>f\left(x\right)$ and $f\left(z\right)>f\left(x\right)$, without loss of generality, assume $f\left(z\right)\geqslant f\left(y\right)$. Then we have for any $t>0$, $f\left(z+t\left(z-y\right)\right)\geq f\left(z\right)>f\left(x\right)$.
We know a set being convex means picking any two points in that set, the points lying inside the two are still in that set. Let's consider another property: picking any two points in a set, all the points outside one of the two points are still in that set. Then the above paragraph means the upper contour set of a convex function has this property. I was wondering whether there is an established name for this property.