Three-body $B^+to p overline p K^+$ and $B^+to p overline p pi^+$ decays are studied using a data sample corresponding to an integrated luminosity of 3.0 $fb^{-1}$ collected by the LHCb experiment in proton-proton collisions at center-of-mass energies of $7$ and $8$ TeV. Evidence of CP violation in the $B^+to p overline p K^+$ decay is found in regions of the phase space, representing the first measurement of this kind for a final state containing baryons. Measurements of the forward-backward asymmetry of the light meson in the $poverline p$ rest frame yield $A_{mathrm{FB}}(p overline p K^+,~m_{poverline p}<2.85mathrm{,Gekern -0.1em V!/}c^2)=0.495pm0.012~(mathrm{stat})pm0.007~(mathrm{syst})$ and $A_{mathrm{FB}}(poverline p pi^+,~m_{poverline p}<2.85mathrm{,Gekern -0.1em V!/}c^2)=-0.409pm0.033~(mathrm{stat})pm0.006~(mathrm{syst})$. In addition, the branching fraction of the decay $B^+tokern 0.1emoverline{kern -0.1emLambda}(1520)p$ is measured to be $mathcal{B}(B^+ to kern 0.1emoverline{kern -0.1emLambda}(1520)p)=(3.15pm0.48~(mathrm{stat})pm0.07~(mathrm{syst})pm0.26~(mathrm{BF}))times 10^{-7}$, where BF denotes the uncertainty on secondary branching fractions.
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