Evangelista gave birth to her son Augie when Francois-Henri Pinault, was dating Hayek. The model claims Pinault, who is heir to a 11.5 billion-dollar family estate, should give their son more than 1,500 dollars-a-day - 807,000 dollars-a-year - for childcare and armed guards for his protection.
The case, which has come to a Manhattan family court as the former couple has been unable to agree on a deal, is expected to last four days.
When she brought the claim in August last year, 46-year-old Evangelista said the money would cover around-the-clock childcare and armed chauffeurs to take care of her son.
“I need to protect him,” the Daily Mail quoted Evangelista as telling Love magazine in February. Evangelista is also demanding Pinault give her son a home similar to the 13 million-dollar California mansion he put in a trust for his and Hayek’s daughter.