----- - Let $X$ be a [[convex set|convex subset]] of a [[metric space]]. > [!proposition] Proposition. ([[strictly convex functions have at most one global min]]) > If $f: X \to \rr$ is a [[strictly convex function]], then $f$ has *at most* one [[global extrema|global min]]. > \ > **Remark.** Together with [[local mins of convex functions are global mins]] we have that any [[strictly convex function]] has at most one minimizer. > > [!proof]- Proof. ([[strictly convex functions have at most one global min]]) > Suppose $f$ has two [[global extrema|global mins]], $\v u ^{*}$ and $\v v ^{*}$. Then in particular $f(\v u ^{*}) \leq f(\v v^{ *})$ and $f(\v u ^{*}) \geq f(\v v^{ *})$, hence $f(\v u ^{*})=f(\v v^{*})$. For all $t \in [0,1)$ we have, since $f$ is [[strictly convex function|strictly convex]], that $\begin{align}f(\v v ^{*}) \leq f(t \v u^* + (1-t)\v v^*) < & tf(\v u ^*) + (1-t)f(\v v^{*}) \\ = & tf(\v u ^{*}) +(1-t)f(\v u ^{*} ) \\ & = f(\v u ^{*}) , \end{align}$ this is a contradiction since $\v u ^{*}$ is a [[global extrema|global extrema]]. $\qedin$ ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```