  We study inclusive di-pion decays using a sample of 108 x 10^6 Upsilon(3S)
events recorded with the BABAR detector. We search for the decay mode
Upsilon(3S) --> pi+ pi- h_b(1P) and find no evidence for the bottomonium
spin-singlet state h_b(1P) in the invariant mass distribution recoiling against
the pi+ pi- system. Assuming the h_b(1P) mass to be 9.900 GeV/c^2, we measure
the upper limit on the branching fraction B[Upsilon(3S) --> pi+ pi- h_b(1P)] <
1.2 x 10^{-4}, at 90% confidence level. We also investigate the chi_{bJ}(2P)
--> pi+ pi- chi_{bJ}(1P), Upsilon(3S) --> pi+ pi- Upsilon(2S), and Upsilon(2S)
--> pi+ pi- Upsilon(1S) di-pion transitions and present an improved measurement
of the branching fraction of the Upsilon(3S) --> pi+ pi- Upsilon(2S) decay and
of the Upsilon(3S)-Upsilon(2S) mass difference.
