▁of ▁as ▁paramet riz ation ▁of ▁the ▁space ▁of ▁ge odes ics . ▁ ▁Valid ity ▁ ▁Under ▁the ▁assumptions ▁that ▁▁▁ ▁is ▁non - tra pping ▁( i . e . ▁ ▁for ▁all ▁) ▁and ▁▁▁ ▁is ▁strictly ▁convex ▁( i . e . ▁the ▁second ▁fundamental ▁form ▁ ▁is ▁positive ▁definite ▁for ▁every ▁), ▁ ▁Sant al ó ' s ▁formula ▁is ▁valid ▁for ▁all ▁. ▁In ▁this ▁case ▁it ▁is ▁equivalent ▁to ▁the ▁following ▁identity ▁of ▁measures : ▁ ▁where ▁ ▁and ▁ ▁is ▁defined ▁by ▁. ▁In ▁particular ▁this ▁implies ▁that ▁the ▁ge odes ic ▁X - ray ▁transform ▁▁ ▁extends ▁to ▁a ▁bounded ▁linear ▁map ▁, ▁where ▁ ▁and ▁thus ▁there ▁is ▁the ▁following , ▁- version ▁of ▁Sant al ó ' s ▁formula : ▁ ▁If ▁the ▁non - tra pping ▁or ▁the ▁convex ity ▁condition ▁from ▁above ▁fail , ▁then ▁there ▁is ▁a ▁set ▁ ▁of ▁positive ▁measure , ▁such ▁that ▁the ▁ge odes ics ▁emer ging ▁from ▁ ▁either ▁fail ▁to ▁hit ▁the ▁boundary ▁of ▁ ▁or ▁hit ▁it ▁non - trans vers ely . ▁In ▁this ▁case ▁Sant al ó ' s ▁formula ▁only ▁remains ▁true ▁for ▁functions ▁with ▁support ▁dis joint ▁from ▁this ▁exception al ▁set ▁. ▁ ▁Proof ▁ ▁The ▁following ▁proof ▁is ▁taken ▁from ▁[ , ▁Lemma ▁ 3 . 3 ], ▁adapted ▁to ▁the ▁( sim pler ) ▁setting ▁when ▁conditions ▁ 1 ) ▁and ▁ 2 ) ▁from ▁above ▁are ▁true . ▁Sant al ó ' s ▁formula