Noteworthy Uses:: *[[Noteworthy Uses]]* Proved By:: [[triangle inequality]] Intuition:: *[[Intuition]]* ----- > [!proposition] Proposition. ([[vector norm is convex]]) > The [[vector]] [[norm]] is a [[convex function]]. > [!proof]- Proof. ([[vector norm is convex]]) > Let $V$ be a [[norm|normed vector space]]; let $\v u, \v v \in V$. By the triangle inequality, for all $t \in [0,1)$ we have $\|t\v u + (1-t)\v v\| \leq \|t \v u\| + \|(1-t)\v v\|= t \|\v u\| + (1-t)\|\v v\|,$ which is the definition of [[convex function|convexity]]. ----- #### ---- #### 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 > ```