Police have issued a new appeal to trace a couple who have been missing with their newborn baby, saying they believe they may be running low on cash.
It is believed Constance Marten, Mark Gordon and their baby have been camping out in the Sussex countryside.
Officers are concerned for the wellbeing of the baby, who spent the first month of their life "exposed to the elements".
The Metropolitan Police said the baby needs medical attention and they are asking the couple to get in touch to confirm if they are okay.
After being missing for a month, investigators also believe the family may be "running low on cash" and want to ensure the baby is "safe and warm".
Addressing the pair directly Det Supt Basford said: "Constance and Mark, your baby has spent the first month of its short life exposed to the elements when it should be safe and warm and, most importantly, seen by medics.
"After a month you must be running low on cash. Please pick up the phone and let us know, at the very least, that you are okay. We are ready to come to you and see that you and the baby get medical attention."
The family were dropped off in a taxi at 4.56am on January 8 outside the entrance to the port in Newhaven in East Sussex.
The Met said they walked to where the A259 crosses over the B2109 and were seen sheltering form the rain, under the overpass, at about 6am.
At 6.15am, they walked along Cantercrow Hill into the fields beyond - carrying bags and their blue tent.
For this reason, officers believe they have been camping in the area but because a month has passed in the meantime, they may have moved on "some considerable distance".
The Met said the couple no longer have a pushchair they were previously seen with after they dumped it on January 7.
Detective Superintendent Lewis Basford of the Met's East Area Command said: "Previous appeals have resulted in around 300 calls from members of the public, but sadly they have not led to the couple and their baby being found.
"While we are very grateful to everyone who has already called, we still need to hear from anyone who has seen the couple since Sunday, 8 January and we are offering a reward of up to £10,000 for any information that leads to the family being found.
"I am appealing in particular to people who may have spent time in the countryside near Newhaven and elsewhere in Sussex to think carefully as to whether they may have seen Constance and Mark over the past four weeks.
"Given the time that's passed they may have travelled on, so I also need to hear from anyone who may have seen them further afield."
He added: "For a month now Mark, Constance and her new-born have been continually on the move in sub-zero temperatures with no medical attention and we are increasingly concerned for their wellbeing.
"We know that the baby was still alive on 8 January and finding the baby remains our top priority. Maybe you have information but were reluctant to come forward to help us find the family.
"It does not matter why you did not speak to us earlier, what matters is doing the right thing now for the good of this highly vulnerable infant.
"If you have seen the family, if you gave them directions, if they got into your taxi or came into your shop to buy food, drink or nappies - no matter how small a detail it may seem - please do get in touch.
"Please also be assured that if they have paid you to provide a service, a lift, somewhere to stay or something else, you are not in any sort of trouble, we just need you to tell us what you know so that we can find them and make them safe."
Gordon is a convicted rapist who was jailed in America in 1990 for the rape of a woman in Florida.
After serving his lengthy sentence he was deported to the UK in 2010 and placed on the Sex Offenders Register.
It is thought the 48-year-old has been in a relationship with Ms Marten, 35, since 2016, who comes from a wealthy aristocratic family with links to the royals.
Her grandmother was reportedly a playmate of Princess Margaret, while her father Napier Marten was a page to Queen Elizabeth.
Anyone who has information on the family’s whereabouts should call the incident room on 020 7175 0785.
Alternatively, information can be reported 100 percent anonymously to the independent charity Crimestoppers on 0800 555 111. They never ask for personal details and they do not trace your device.
