Do you have a citation for this? I don't see such a sense among the glosses given here:
http://www.perseus.tufts.edu/hopper/morph?l=sinus&la=la&can=...The closest one appears to be II.A.2.d:
> A hiding-place, place of concealment: ut in sinu gaudeant, gloriose loqui desinunt, qs. in their bosoms (or, as we say, in their sleeve), i. e. in secret, Cic. Tusc. 3, 21, 51; “so of secret joy,” Tib. 4, 13, 8: “in tacito cohibe gaudia clausa sinu,” Prop. 2, 25 (3, 20), 30; Sen. Ep. 105, 3; cf. “also: plaudere in sinum,” Tert. Pudic. 6: suum potius cubiculum ac sinum offerre contegendis quae, etc., the secrecy or concealment of her bed-chamber, Tac. A. 13, 13: “abditis pecuniis per occultos aut ambitiosos sinus,” i. e. in hiding places offered by obscurity or by high rank, id. H. 2, 92.—
But those examples make it pretty clear that the sense of sinus is secrecy, not emptiness. Presumably this comes from the idea that you can hold ideas and opinions within your heart (in your breast, sinus) and no one else can see them.