  Locating the gamma-ray emission sites in blazar jets is a long-standing and
highly controversial issue. We investigate jointly several constraints on the
distance scale r and Lorentz factor Gamma of the gamma-ray emitting regions in
luminous blazars (primarily flat spectrum radio quasars, FSRQs). Working in the
framework of one-zone external radiation Comptonization (ERC) models, we
perform a parameter space study for several representative cases of actual
gamma-ray flares in their multiwavelength context. We find a particularly
useful combination of three constraints: from an upper limit on the collimation
parameter Gamma*theta <~ 1, from an upper limit on the synchrotron self-Compton
(SSC) luminosity L_SSC <~ L_X, and from an upper limit on the efficient cooling
photon energy E_cool,obs <~ 100 MeV. These three constraints are particularly
strong for sources with low accretion disk luminosity L_d. The commonly used
intrinsic pair-production opacity constraint on Gamma is usually much weaker
than the SSC constraint. The SSC and cooling constraints provide a robust lower
limit on the collimation parameter Gamma*theta >~ 0.1 - 0.7. Typical values of
r corresponding to moderate values of Gamma ~ 20 are in the range 0.1 - 1 pc,
and are determined primarily by the observed variability time scale t_var,obs.
Alternative scenarios motivated by the observed gamma-ray/mm connection, in
which gamma-ray flares of t_var,obs ~ a few days are located at r ~ 10 pc, are
in conflict with both the SSC and cooling constraints. Moreover, we use a
simple light travel time argument to point out that the gamma-ray/mm connection
does not provide a significant constraint on the location of gamma-ray flares.
We argue that spine-sheath models of the jet structure do not offer a plausible
alternative to external radiation fields at large distances, however, an
extended broad-line region is an idea worth exploring.
