  Using the fact that flat space with a boundary is related by a Weyl
transformation to anti-de Sitter (AdS) space, one may study observables in
boundary conformal field theory (BCFT) by placing a CFT in AdS. In addition to
correlation functions of local operators, a quantity of interest is the free
energy of the CFT computed on the AdS space with hyperbolic ball metric, i.e.
with a spherical boundary. It is natural to expect that the AdS free energy can
be used to define a quantity that decreases under boundary renormalization
group (RG) flows. We test this idea by discussing in detail the case of the
large $N$ critical $O(N)$ model in general dimension $d$, as well as its
perturbative descriptions in the epsilon-expansion. Using the AdS approach, we
recover the various known boundary critical behaviors of the model, and we
compute the free energy for each boundary fixed point, finding results which
are consistent with the conjectured $F$-theorem in a continuous range of
dimensions. Finally, we also use the AdS setup to compute correlation functions
and extract some of the BCFT data. In particular, we show that using the bulk
equations of motion, in conjunction with crossing symmetry, gives an efficient
way to constrain bulk two-point functions and extract anomalous dimensions of
boundary operators.
