On rigid $q$-plurisubharmonic functions and $q$-pseudoconvex tube domains in $\mathbb{C}^n$

In the spirit of Lelong and Bochner, we show that an upper semi-continuous function defined on a open tube set $\Omega=\omega + i\mathbb{R}^n$ in $\mathbb{C}^n$, where $\omega$ is an open set in $\mathbb{R}^n$, and which is invariant in its imaginary part, is $q$-plurisubharmonic on $\Omega$ (in the sense of Hunt and Murray) if and only if it is real $q$-convex on $\omega$, i.e., it admits the local maximum property with respect to affine linear functions on real $(q+1)$-dimensional affine subspaces. From this, we conclude that, for $a>0$, the set $\omega+i(-a,a)^n$ is $q$-pseudoconvex in $\mathbb{C}^n$ if and only if $\omega$ is a real $q$-convex set in $\mathbb{R}^n$, i.e., $\omega$ admits a real $q$-convex exhaustion function on $\omega$. We apply these results to complements of graphs of affine linear maps and to Reinhardt domains.