PomeLemma 19.7.21.DCB8E8464FB6 Let $S,S'\in \mathcal{S}(C)$. Then
\[ |\operatorname{vol}S-\operatorname{vol}S'|\leq d_{\mathrm{sg}}(S,S'). \]
Lemma 19.7.21DCB8E8464FB6
Proof. Choose a sufficiently divisible positive integer $a$. Taking the limsup in the definition of $d_{\mathrm{sg}}$ along $k=ma$ gives
\[ d_{\mathrm{sg}}(S,S')\geq \operatorname{vol}S+\operatorname{vol}S'-2\operatorname{vol}(S\cap S'). \]
It follows that $\operatorname{vol}S-\operatorname{vol}S'\leq d_{\mathrm{sg}}(S,S')$ and $\operatorname{vol}S'-\operatorname{vol}S\leq d_{\mathrm{sg}}(S,S')$. $\square$
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