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Let $u \in X$, let $G$ be a Eucildean space and let $L:X \to G$ be a linear operator. Show that convexity of the following function:

$h(x)=\begin{cases} -\sqrt{|\langle x, u \rangle|^2 - \lVert Lx \rVert^2}, & \text{if } \langle x, u \rangle > 0 \text{ and } \lVert Lx \rVert < \langle x, u \rangle \\ +\infty, & \text{otherwise} \end{cases}$

My attempt:

$L$ is convex, and $||.||^2$ is convex and increasing, thus $||Lx||^2$ is convex. Thus, $-||Lx||^2$ is concave.
$f(x)=\langle x, u \rangle$ is linear(convex). $h(x)=|.|^2$ is convex and increasing, thus I get $|\langle x,u \rangle|^2$ to be convex. But, I am unable to say something about the sum of a convex and concave function.

lone_wolf
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1 Answers1

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hint: It looks like you are trying to use a result along the lines of "convex and increasing composed with [something] yields a convex function" which should be the right direction. However, instead of your current choice of $h=|\cdot|^2$, something else may yield the result.

Also, since convexity is a statement for scalar-valued functions and $L$ is a generic linear operator which may be vector-valued, the statement "$L$ is convex" should be changed too. However, for your class you may be able to use the fact that a convex function composed with a linear operator remains convex.

Zim
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