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Abstract For the first time, the strong phase difference between D 0 and D¯0→π+π−π+π− $$ {\overline{D}}^0\to {\pi}^{+}{\pi}^{-}{\pi}^{+}{\pi}^{-} $$ amplitudes is determined in bins of the decay phase space. The measurement uses 818 pb−1 of e + e − collision data that is taken at the ψ(3770) resonance and collected by the CLEO-c experiment. The measurement is important for the determination of the CP -violating phase γ in B ± → DK ± (and similar) decays, where the D meson (which represents a superposition of D 0 and D¯0 $$ {\overline{D}}^0 $$) subsequently decays to π + π − π + π −. To obtain optimal sensitivity to γ, the phase space of the D → π + π − π + π − decay is divided into bins based on a recent amplitude model of the decay. Although an amplitude model is used to define the bins, the measurements obtained are model-independent. The CP -even fraction of the D → π + π − π + π − decay is determined to be F +4π  = 0.769 ± 0.021 ± 0.010, where the uncertainties are statistical and systematic, respectively. Using simulated B ± → DK ±, D → π + π − π + π − decays, it is estimated that by the end of the current LHC run, the LHCb experiment could determine γ from this decay mode with an uncertainty of (±10 ± 7)°, where the first uncertainty is statistical based on estimated LHCb event yields, and the second is due to the uncertainties on the parameters determined in this paper.