In Boyd and Vandenberghe's "Convex Optimization":
The $\alpha$-sublevel set set of a function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is defined as
$$C_\alpha=\{x \in \mathbf{dom} f|f(x)\leq\alpha\}.$$
After stating that "sublevel sets of a convex function are convex, for any value of $\alpha$", the authors say that the converse is not true, and provide $f(x)=-e^{x}$ as an example of a strictly concave function in $\mathbb{R}$ whose sublevel sets are convex for any value of $\alpha$.
Although I was able to check this example in an analitical fashion, I am still wondering if there is a geometric (and more intuitive) interpretation of the $\alpha$-sublevel set of a function that helps the evaluation of this set's convexity properties.
(As an example, remember that, geometrically, a function $f$ is said to be convex if the cord/line segment between $(x,f(x))$ and $(y,f(y))$ lies above the graph $f$.)
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$\begingroup$The $\alpha$-sublevel set is the set of all $x$ where $f(x) \le \alpha$, i.e. the point on the graph corresponding to $x$ lies on or below the horizontal plane (or line in the case $n=1$) $y=\alpha$.
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