[iso1631]

You can certainly increase density while maintaining green spaces. Westminster (A London borough) has enormous open areas -- Hyde Park, Green Park, Regents Park and St James Park - taking almost half the entire borough's area.

It still has a population of 25,000 per square mile. London as a whole has a density of 14,600. You could add another 6 million people in London while maintaining a density which would allow a third of London to be open green areas.

[HDThoreaun]

West London is also the most expensive part, completely unaffordable for 95% of people. It's not a coincidence that the best amenities are there. Certainly not clear that you can have high amenity areas that are also affordable.

[iso1631]

it's expensive because it's mostly nice (although the cheapest place I ever lived in was near west kensington tube station, just on the wrong side of the tracks, but about a mile from kensington gardens.

It's quite clear that you can have housing for 15 million people in the area of Greater London with the density and green space of Westminster.

Does that require grand architecture? Sure. Will it happen? Nope. But it's not a density problem, it's an allocation problem.

Instead great swathes of London are built over with high density semi and terrace housing with a tiny bit of poor green space per person (usually paved over to store a car which sits still 23 hours a day or more)

[agd]

Yep, it's possible, but hard in practice for lots of places.

London has historical parks/commons which are fantastic, but for cities which don't have many large parks, it's hard to retrofit them.