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# Table 1 Verifying that the corner points of inner and outer bounds coincide

Case $$d_{2}' = \text {min}\left \{d_{2}^{\mathsf {max}}, d_{\mathsf {sum}}^{\mathsf {max}} - d_{1}^{\mathsf { max}}\right \}$$
$${L}_{T_{1}}|\Psi _{T_{11}}| \geq {L}_{R_{1}}|\Psi _{R_{11}}|,$$
$$\text {min}\{d_{2}^{\mathsf {max}},\ 2{L}_{T_{2}}|\Psi _{T_{22}}\cup \Psi _{T_{12}}|$$
$${\phantom {0000}}-2{L}_{R_{1}}|\Psi _{R_{11}}\cap \Psi _{R_{12}}| \}$$
$${L}_{T_{2}}|\Psi _{T_{12}}| \geq {L}_{R_{1}}|\Psi _{R_{12}}|$$
$${L}_{T_{1}}|\Psi _{T_{11}}| \geq {L}_{R_{1}}|\Psi _{R_{11}}|,$$
$$\text {min}\{d_{2}^{\mathsf {max}},\ 2{L}_{T_{2}}|\Psi _{T_{22}}\setminus \Psi _{T_{12}}|$$
$$+ 2{L}_{R_{1}} |\Psi _{R_{12}}\setminus \Psi _{R_{11}}|\}$$
$${L}_{T_{2}}|\Psi _{T_{12}}| < {L}_{R_{1}}|\Psi _{R_{12}}|$$
$${L}_{T_{1}}|\Psi _{T_{11}}| < {L}_{R_{1}}|\Psi _{R_{11}}|,$$
$$\text {min}\{d_{2}^{\mathsf {max}},\ 2{L}_{T_{2}}|\Psi _{T_{22}}\cup \Psi _{T_{12}}|$$
$$+ 2{L}_{R_{1}}|\Psi _{R_{11}}\setminus \Psi _{R_{12}}| - 2{L}_{T_{1}}|\Psi _{T_{11}}| \}$$
$${L}_{T_{2}}|\Psi _{T_{12}}| \geq {L}_{R_{1}}|\Psi _{R_{12}}|$$
$${L}_{T_{1}}|\Psi _{T_{11}}| < {L}_{R_{1}}|\Psi _{R_{11}}|,$$
$$\text {min}\{d_{2}^{\mathsf {max}},\ 2{L}_{T_{2}}|\Psi _{T_{22}}\setminus \Psi _{T_{12}}|$$
$$+ 2{L}_{R_{1}}|\Psi _{R_{11}}\cup \Psi _{R_{12}}| - 2{L}_{T_{1}}|\Psi _{T_{11}}| \}$$
$${L}_{T_{2}}|\Psi _{T_{12}}| < {L}_{R_{1}}|\Psi _{R_{12}}|$$