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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 $f$.)

asimoes
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1 Answers1

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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 $n=1$) $y=\alpha$.

Robert Israel
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  • In that case, the $\alpha$-sublevel set of $f(x)=−e^x$ would be $x\in[\log(−\alpha),+\infty[$ (if $\alpha \geq 0$), then $x\in \mathbb{R}$ instead), which is a convex set because our single variable $x$ can take any value contained in that continuous interval (i.e. $\alpha$-sublevel set describes a line segment). Is this correct? – asimoes Feb 20 '16 at 01:25