  We initiate the study of $m$-subharmonic functions with respect to a
semipositive $(1,1)$-form in Euclidean domains, providing a significant element
in understanding geodesics within the context of complex Hessian equations.
Based on the foundational Perron envelope construction, we prove a
decomposition of $m$-subharmonic solutions, and a general comparison principle
that effectively manages singular Hessian measures. Additionally, we establish
a rooftop equality and an analogue of the Kiselman minimum principle, which are
crucial ingredients in establishing a criterion for geodesic connectivity among
$m$-subharmonic functions, expressed in terms of their asymptotic envelopes.
