Use of a possessive noun in place of a location

My friend and I have been arguing about the following sentence:

"I left my trainers at my relatives’ in London"

He believes this is correct as colloquially it is obvious that he has left his trainers at the home of his relatives. But I still think that this sentence is incomplete and should read:

"I left my trainers at my relatives’ home in London"

Could someone please tell me who is correct (if anyone) and explain why?

Solution 1:

The use of the Saxon genitive with the head noun dropped, to describe locations, is quite common in English.

• McDonald’s
• We’ve run out of bread. I’m off to the baker’s.
• Go to the doctor’s if you’re not feeling well.
• He has been to London quite a few times, but has never actually seen St Paul’s.
• She stayed at her uncle’s for over a month.
• I left my trainers at my relatives’ in London.

The use of this location genitive has indeed a colloquial ring to it, but only slightly so.