Properties:: strictly convex is convex (obviously), [[strictly convex functions have at most one global min]] Sufficiencies:: [[C2 function is strictly convex if its hessian is everywhere positive definite]] Equivalences:: [[convex iff graph lies above all its tangents|strictly convex if graph lies strictly above all its tangents]] ---- - Let $X$ be a [[convex set|convex subset]] of a [[metric space]]; - Let $f: X \to \rr$. > [!definition] Definition. ([[strictly convex function]]) > We call $f$ **strictly convex** if $f\big(t \v u + (1-t)\v v\big) < tf(\v u) + (1-t)f(\v v)$ for all $\v u, \v v \in X$ and $t \in [0,1)$. ---- #### ---- #### 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 > ```