  We study codimension one branes, i.e. p-branes in (p+2)-dimensions, in the
superembedding approach for the cases where the worldvolume superspace is
embedded in a minimal target superspace with half supersymmetry breaking. This
singles out the cases p=1,2,3,5,9. For p=3,5,9 the superembedding geometry
naturally involves a fundamental super 2-form potential on the worldvolume
whose generalised field strength obeys a constraint deducible from considering
an open supermembrane ending on the p-brane. This constraint, together with the
embedding constraint, puts the system on-shell for p=5 but overconstrains the
9-brane in D=11 such that the Goldstone superfield is frozen. For p=3 these two
constraints give rise to an off-shell linear multiplet on the worldvolume. An
alternative formulation of this case is given in which the linear multiplet is
dualised to an off-shell scalar multiplet. Actions are constructed for both
cases and are shown to give equivalent equations of motion. After gauge fixing
a local Sp(1) symmetry associated with shifts in the Sp(1)_R Goldstone modes,
we find that the auxiliary fields in the scalar multiplet parametrise a
two-sphere. For completeness we also discuss briefly the cases p=1,2 where the
equations of motion (for off-shell multiplets) are obtained from an action
principle.
