I have a polynomial that I know to be convex. If I homogenize the polynomial, is the resulting homogeneous polynomial also convex? I know that the perspective of a convex function is convex, but cannot find a citation for the homogenization of a convex function.

The function whose convexity I would like to show in the variables $(a,p_1,\dots,p_N)$ for $a > 0$ and under some bound on the magnitudes of the $p_n$'s (I suspect the bound will be $\vert p_n \vert < a$) is

$a^2 \prod_{n=1}^N \left(1 + \vert p_n \vert^2/a^2\right)$.

The function $\prod_{n=1}^N \left(1 + \vert p_n \vert^2\right)$ is log-convex and therefore convex in the $p_n$'s, but I am having difficulty showing that the original function is convex. However, the original function is a homogenized version of this convex function.

Thank you!