Dalitz plot binning optimization for the model-independent measurement of the CKM angle $β$ with $B^0 \to D π^+ π^-$, $D \to K_S^0 π^+ π^-$ decays

A model-independent measurement of the CKM angle $β$ can be performed using a time-dependent analysis of $B^0 \to D π^+ π^-$, $D \to K_S^0 π^+ π^-$ decays, where the Dalitz plots of both the $B$ and $D$ decays are divided into bins. The choice of binning scheme for the $B^0 \to D π^+ π^-$ Dalitz plo