Locating the gamma-ray emission sites in blazar jets is a long standing and highly controversial issue. We jointly investigate several constraints on the distance scale r and Lorentz factor Γ of the gamma-ray emitting regions in luminous blazars (primarily flat spectrum radio quasars). Working in the framework of one-zone external radiation Comptonization 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 Γθ 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 Γ is usually much weaker than the SSC constraint. The SSC and cooling constraints provide a robust lower limit on the collimation parameter Γθ 0.1-0.7. Typical values of r corresponding to moderate values of Γ ~ 20 are in the range 0.1-1 pc, and are determined primarily by the observed variability timescale t var, obs. Alternative scenarios motivated by the observed gamma-ray/millimeter 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/millimeter 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. We propose that the most definite additional constraint could be provided by determination of the synchrotron self-absorption frequency for correlated synchrotron and gamma-ray flares.
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