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Consider a convex function $f: U \to \mathbb{R}$ where $U \subset \mathbb{R}^{m}$ is convex compact. Such convex function may have multiple minimizers, i.e., $\arg \min\limits_{u \in U}f$ has more than one element.

Given $\epsilon > 0$, I'd like to find a convex function $g: U \to \mathbb{R}$ such that $\lvert \lvert f - g\rvert \rvert_{\infty} < \epsilon$ and $g$ has a unique minimizer.

My insight is, choose an arbitrary minimizer $u^{*}$ of the function $f$, and let $z$ be the corresponding point $(u^{*}, \min\limits_{u \in U}f(u) - \delta)$ with positive constant $\delta > 0$.

And consider the convex hull of the epigraph of $f$ and $z$, namely, $Z$. There (probably) exists a $\delta > 0$ such that the function $g$ corresponding to the epigraph $Z$ will satisfy the requirement. Is there something like this, or, is my insight correct and provable?

Rodrigo de Azevedo
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Jinrae Kim
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    Add to it some strictly convex continuous function on $U$ multiplied by some number $\varepsilon$. The sum is strictly convex, hence has a unique minimum. The difference is uniformly bounded by the boundedness theorem, and can be made arbitrarily small using $\varepsilon$. – Conifold Nov 05 '22 at 03:59
  • Ah, you're right! It is a very straightforward and easy solution... Why couldn't I come up with this? Thank you! The answer is enough, but can I ask you a further question? "How can I properly construct such a strictly convex continuous function in general?" I mean, I know we can find such a function, but I'm not sure how I can "construct" it in general. – Jinrae Kim Nov 05 '22 at 04:04
  • The standard choice is the Euclidean square-norm, the sum of the squares of coordinates, see convex regularization. – Conifold Nov 05 '22 at 04:11
  • Ok, I think this should be fine: for all elements in $\partial U$, give a value $0$. Also, pick an element $v$ from $\arg \min_{u \in U} f$, and give a value $-\epsilon$. And make a convex hull of ${(u, 0): u \in \partial U} \cup {(v, -\epsilon)}$, and retrieve a corresponding function $g$ by regarding the convex hull as the epigraph. – Jinrae Kim Nov 05 '22 at 04:12
  • Just take $g(u)=f(u)+\epsilon |u-u^|^2$ where $\epsilon>0$ is sufficiently small ($U$ is compact) and $u^$ is a minimiser. – copper.hat Nov 05 '22 at 04:19
  • I feel like I'm stupid... yeah, your answer is much easy and clear! – Jinrae Kim Nov 05 '22 at 04:31

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