Consider a function $f:\Theta \subset \mathbb{R}\rightarrow \mathbb{R}$. Let $\Theta$ be a convex and open set and $f$ a strictly convex function.

If the function has a global minimum, is it unique? I'm confused by the fact that $\Theta$ is not assumed to be compact. Is this a problem for guaranteeing uniqueness?

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    $\begingroup$ Strict convexity should be enough: what if $f$ has a global min attained at two distinct points $x_1$ and $x_2$? how about the value of $f$ on the line segment $[x_1,x_2]$? $\endgroup$ – gniourf_gniourf Feb 15 '16 at 19:25

Strict convexity is enough: assume that $f$ has a global min $m$ that is attained at two distinct points $x_1$ and $x_2$. Then, since $f$ is convex: $$\forall t\in[0,1],\ f\bigl(tx_1+(1-t)x_2\bigr)\leq tf(x_1)+(1-t)f(x_2)=tm+(1-t)m=m.$$ Since $m$ is a global min of $f$ we must also have $$\forall t\in[0,1],\ m\leq f\bigl(tx_1+(1-t)x_2\bigr),$$ hence $$\forall t\in(0,1),\ f\bigl(tx_1+(1-t)x_2\bigr)=m=tf(x_1)+(1-t)f(x_2)$$ which is impossible since $f$ is strictly convex.

Hence, if $f$ has a global minimum, this global minimum is attained at only one point.


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