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so it seems that $-log(x)$ is strictly convex but not strongly convex. But what if I define a function, say, $-log(log(log(x)))$ ? and for the sake of the argument, the domain is such that you wouldn't be finding log of negative numbers, thus only dealing with real numbers.

The reason being is that the hessian for $-log(a^{T}X)$ for given vector $a$, is not invertible, but was wondering if making it nested with logs would help make it invertible

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  • $\begingroup$ Can you restrict the domain of $x$ on the positive side? That is to say, can you guarantee that $x\leq M$ for some $M>0$? If so then you do have strong convexity on that restricted domain. $\endgroup$ May 3 '18 at 16:12
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It would not help. The reason that $-\log(x)$ is not strongly convex is that it gets "too flat" as it tends to infinity; a strongly convex function has some positive amount of "curvature" that it has at each point in its domain. Nested logs are "even flatter" than ordinary logarithms, and thus certainly will not be strongly convex/concave either.

Because you are talking about twice differentiable functions on the reals, you can check this by hand relatively easily: just compute the second derivative of whatever nested-log function you're interested in and note that it tends to 0 as $x$ tends to infinity. For the function to be strongly convex, it would have to stay greater than some $\epsilon > 0$.

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