Suppose we have the optimal value function as: \begin{align} v(x) = \max_{y\geq 0}\,&c^\top y\\ s.t. \,&g(x,y) \leq 0. \end{align} If we know $g(x,y)$ is linear in $x$, but nonconvex in $y$, is $v(x)$ a convex function? If no, under what conditions on $g(x,y)$ such that $v(x)$ is convex in $x$? (I know for sure that if $g(x,y)$ is also linear in $y$, $v(x)$ is convex.)
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