Line connection function in convex function is convex

by Christian.gruener   Last Updated July 12, 2019 09:20 AM - source

$\newcommand\R{\mathbb R} \newcommand\x{\mathbf x} \newcommand\y{\mathbf y} $Let $\gamma\colon\R\to\R^n$ be the curve defined by $t\mapsto \x+t(\y-\x)$, which is a straight line through $\gamma(0)=\x$ and $\gamma(1)=\y$. Further define $\widetilde V=V\circ \gamma:\R\to\R$.

Check that $\widetilde V$ is convex given that $V$ is convex. I think this is suppose to be just using the definition and readjustment of the terms, but I am not able to do it. Any tips are welcome.



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