Is a function that is convex wrt 3 subsets of variables also a convex function?

Question

Function $f(\vec{x}, \vec{y}, \vec{z})$ is convex if I fix any two of the vector variables.

For example, $g(\vec{x}) = f(\vec{x}, \vec{1}, \vec{2})$ is convex.

Is $f$ convex?

Can I find the global optimum of $f$ by:

1. Convex optimization over $\vec{x}$ and keep $\vec{y}$ and $\vec{z}$ fixed.
2. Convex optimization over $\vec{y}$ and keep $\vec{z}$ and $\vec{x}$ fixed.
3. Convex optimization over $\vec{z}$ and keep $\vec{x}$ and $\vec{y}$ fixed.
4. repeat.

Answer 1

$xyz$ is not convex, but trivially convex (and concave) if you fix any two variables.