Time-dependent analysis ofB0→KS0π−π+γdecays and studies of theK+π−π+system inB+→K+π−π+γdecays

We measure the time-dependent CP asymmetry in the radiative-penguin decay B-0 -> K-S(0)pi(-)pi(+)gamma, using a sample of 471 x 10(6) Upsilon(4S) -> B (B) over bar events recorded with the BABAR detector at the PEP-II e(+)(e) over tilde (-) storage ring at SLAC. Using events with m(K pi pi) < 1.8 GeV/c(2), we measure the branching fractions of B+ -> K+pi(-)pi(+)gamma and B-0 -> K-S(0)pi(-)pi(+)gamma, the branching fractions of the kaonic resonances decaying to K+pi(-)pi(+)gamma, as well as the overall branching fractions of the B+ -> rho K-0(+.)gamma, B+ -> K*(0)pi(+)gamma. and S-wave B+ -> (K pi)(0)(*0) pi(+)gamma components. For events from the rho mass band, we measure the CP-violating parameters SKS0 pi+pi-gamma = 0.14 +/- 0.25 +/- 0.03 and CKS0 pi+pi-gamma = -0.39 +/- 0.20(-0.02)(+0.003), where the first uncertainties are statistical and the second are systematic. We extract from this measurement the time-dependent CP asymmetry related to the CP eigenstate rho K-0(S)0 and obtain S-KS(0) = -0.18 +/- 0.32(-0.05)(+0.06), which provides information on the photon polarization in the underlying b -> s gamma transition.