Suppose I have a function $f(x,y)$ from $\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ that is convex in both $x$ and $y$. Set

$g(y) = \min_{x} f(x,y)$

What I would like is for $g(y)$ to be Lipschitz:

$|g(y) - g(y')| \le c \cdot \| y - y' \|$

Unfortunately, $f(x,y)$ may have a very poor Lipschitz constant for general $x$. Are there general conditions on $f$ for which the minima are Lipschitz?

Alternatively, when can we say the minimizer $x^{\ast}(y) = \arg \min_x f(x,y)$ is Lipschitz in $y$?

I've tried looking in a few convex optimization books for answers, but no luck.