  We study the geometry of $m$-regular domains within the
Caffarelli-Nirenberg-Spruck model in terms of barrier functions, envelopes,
exhaustion functions, and Jensen measures. We prove among other things that
every $m$-hyperconvex domain admits an exhaustion function that is negative,
smooth, strictly $m$-subharmonic, and has bounded $m$-Hessian measure.
