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On Page 75 of Boyd's Convex Optimization:

A function can have all its sublevel sets convex but not be a convex function. For example,$ f(x)=-e^x$ is not convex on $R$ but all its sublevel sets are convex.

I don not quite understand that. How to tell the sublevel of $ f(x)=-e^x$ are convex? I think they are all concave as the $f(x)$ itself.

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    The set described by the inequality $-e^x \leq a$, when $a<0$, is exactly equivalent to the set described by $x \geq \log-a$. – 2017-02-12
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    @MichaelGrant Yeah, but isn't it concave? – 2017-02-12
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    You are confusing convex/concave functions and convex sets. There is no such thing as a concave set in the context of convex optimization. – 2017-02-12
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    @MichaelGrant Oh, You are right! – 2017-02-12

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