Optimal binning of $D\rightarrow K_{\mathrm S}^0\pi^+\pi^-$ and $D\rightarrow K_{\mathrm S}^0K^+K^-$ phase space for experimental measurements
New binning schemes for the Dalitz plots of $D \rightarrow K^0_{\mathrm S}\pi^+\pi^-$ and $D \rightarrow K^0_{\mathrm S} K^+ K^-$ decays are presented. These are determined using a new figure of merit for optimisation that better represents the sensitivity to $\gamma$ than previous metrics. Further augmentation comes from including consideration of degeneracy with other physics parameters determined simultaneously and a more accurate treatment of relevant backgrounds. The overall expected improvement in the precision of $\gamma$ measurements using $B^\pm\to DK^\pm$ decays, followed by $D \rightarrow K^0_{\mathrm S}\pi^+\pi^-$, is estimated at around 5$\%$. In addition, the first dedicated optimisation is performed for the observables relevant to charm mixing, $x_{CP}$ and $\Delta x$, in $D \rightarrow K^0_{\mathrm S}\pi^+\pi^-$ decays. This procedure accounts for both statistical sensitivity and the migration of events between bins due to detector resolution. The resulting binning scheme leads to an estimated gain of approximately 20$\%$ in statistical sensitivity, while maintaining the bias due to bin migration at a level comparable to the currently used scheme.