Convex function, Epigraph, Sublevel set The following graph represents the sublevel set $ S_{-1} = \\{(x,y):f(x,y)\le-1\\} $ (this is not an epigraph!). Is the function $ f(x,y)=-2e^{x} +y$ convex? Is it quasiconvex? Explain based on the graph. ![enter image description here]( I know that the difference between an epigraph and a sublevel set is that an epigraph like the function is part of $\Bbb R^3$ and a sublevel set part of $\Bbb R^2$. $ -2e^{x} +y\le-1 $ $ y\le2e^{x}-1 $ So, this is the graph of $ -2e^{x} +y=-1 $ The represented sublevel set is not convex and the function is neither convex. Is that correct? Can I say that the function is not convex, because if $f$ is a convex function $\Rightarrow$ it's level sets are convex? This not here the case. I know that if: 1. $f$ is convex $\Leftarrow\Rightarrow$ $epi f$ is a convex level set 2. $f$ is convex $\Rightarrow$ all $S_{k}$ sublevel sets must be convex.

The sublevel sets of a convex function are convex. The shaded region is the intersection of a sublevel set with a convex set; it is not convex, therefore the sublevel set is not convex, therefore the function is not convex.