The jet collimation profile at high resolution in BL Lacertae
  Controversial studies on the jet collimation profile of BL Lacertae (BL Lac),
the eponymous blazar of BL Lac objects class, complicate the scenario in this
already puzzling class of objects. Understanding the jet geometry, in
connection with the jet kinematics and the physical conditions in the
surrounding medium, is fundamental to better constrain the formation,
acceleration and collimation mechanisms in extragalactic jets. With the aim of
investigating the jet geometry in the innermost regions of the jet of BL Lac,
and solving the controversy, we explore the radio jet in this source, using
high resolution millimeter-wave VLBI data. We collect 86GHz GMVA and 43GHz VLBA
data to obtain stacked images that we use to infer the jet collimation profile
by means of two comparable methods. We analyze the kinematics at 86GHz, and we
discuss it in the context of the jet expansion. Finally we consider a possible
implication of the Bondi sphere in shaping the different expanding region
observed along the jet. We found that the jet in BL Lac expands with an overall
conical geometry. A higher expanding rate region is observed between ~5 and 10
pc (de-projected) from the black hole. Such a region is associated with the
decrease in brightness usually observed in high-frequency VLBI images of BL
Lac. The jet retrieves the original jet expansion around 17 pc, where the
presence of a recollimation shock is supported by both the jet profile and the
15GHz kinematics (MOJAVE survey). The change in the jet expansion profile
occurring at ~5 pc could be associated with a change in the external pressure
profile in correspondence of the Bondi radius (~3.3X10$^5$$R_s$).
