Consider the following function:
$$f(x, y, z) = x^2 + y^2 - z^2$$
I am writing a paper with such a function in which I stated "$f$ is convex only in variables $x$ and $y$". By this I mean that for any constant $a$, $f(x, y, a)$ is a convex function of $x$ and $y$. I figured my meaning would be clear from context, but my advisor told me that he thinks that this language is potentially ambiguous and asked me to consult the literature to make sure my verbiage is consistent with that which other mathematicians and scientists use for such functions.
The problem is that wherever I have looked I can only find the definition for a function being convex as a whole. I have found no reference that gives any kind of standard notation/name for the situation I described. Can anyone shed some light on this?
Perhaps the answer is right in the title of my question. I could say "$f$ is convex with respect to only $x$ and $y$". But this begs the same: is this standard usage of the language/is the meaning clear and unambiguous?
I would write: "For each value of $a$, the function $f(\cdot, \cdot, a)$ is convex" or (very similarly) "For each value of $a$, the function $(x,y) \mapsto f(x,y,a)$ is convex".