  The S-matrix bootstrap is extended to a 1+1d theory with $O(N)$ symmetry and
a boundary in what we call the R-matrix bootstrap since the quantity of
interest is the reflection matrix (R-matrix). Given a bulk S-matrix, the space
of allowed R-matrices is an infinite dimensional convex space from which we
plot a two dimensional section given by a convex domain on a 2d plane. In
certain cases, at the boundary of the domain, we find vertices corresponding to
integrable R-matrices with no free parameters. In other cases, when there is a
one-parameter family of integrable R-matrices, the whole boundary represents
integrable theories. We also consider R-matrices which are analytic in an
extended region beyond the physical cuts, thus forbidding poles (resonances) in
that region. In certain models, this drastically reduces the allowed space of
R-matrices leading to new vertices that again correspond to integrable
theories. We also work out the dual problem, in particular in the case of
extended analyticity, the dual function has cuts on the physical line whenever
unitarity is saturated. For the periodic Yang-Baxter solution that has zero
transmission, we computed the R-matrix initially using the bootstrap and then
derived its previously unknown analytic form.
