# On the duality gap for quasiconvex optimisation problems

This stack exchange question got me thinking about quasiconvex analysis.

Given a compact,convex subset $X\subset \mathbb{R}^n$ and a quasiconvex function $f:X\rightarrow \mathbb{R}$

Define the double Legendre-Fenchel transform of $f$, written $f^{**}$, given by: $$\operatorname{epi} (f^{**})=\operatorname{co}(\operatorname{epi}(f))$$ where $\operatorname{epi}(f)\in X\times\mathbb{R}$ is the epigraph of f, and $\operatorname{co}$ is the convex hull of a set.

Assume $f$ has a unique minimum

I know that bad things can happen here if $X$ is not compact, but for the closed and bounded case, it seems like $f^{**}$ can also only have a single minimum on $X$, and this will also minimize $f$. Does someone have a counter-example or a proof?

My intuition has failed me often enough in the past, so I thought I would ask before making an ass of myself.

EDIT: I decided to provide a (simple) example which is indicative of my intuition on this matter. Consider $f(x)=\sqrt{\left| x\right| }$. If we take the domain to be the entire real axis, then $f^{**}\equiv 0$. But for any closed interval $[a,b]$ , $(f\left| _{[a,b]} \right.)^{**}$ fulfills the qualities that I ask for in my question.

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