  We study the continuum limit of the Benincasa-Dowker-Glaser causal set action
on a causally convex compact region. In particular, we compute the action of a
causal set randomly sprinkled on a small causal diamond in the presence of
arbitrary curvature in various spacetime dimensions. In the continuum limit, we
show that the action admits a finite limit. More importantly, the limit is
composed by an Einstein-Hilbert bulk term as predicted by the
Benincasa-Dowker-Glaser action, and a boundary term exactly proportional to the
codimension-two joint volume. Our calculation provides strong evidence in
support of the conjecture that the Benincasa-Dowker-Glaser action naturally
includes codimension-two boundary terms when evaluated on causally convex
regions.
