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Plans are being drawn up in different parts of the Capital with the aim that as many as possible of people's daily needs – like shops, school, nursery, council services, leisure and even work - are all available within 20 minutes' walk of their homes.
"The 20 minute neighbourhood policy is now getting to a stage where we can start thinking about implementing it," said Councillor McVey.
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"We can start real conversations about what buildings we are talking about, what services, what communities. It's a really exciting point."
The idea of making as much as possible available locally fits with trying to cut car use, boost poorer communities and support local businesses in the wake of the pandemic.
"This is a policy where the climate agenda and the poverty reduction agenda come together," said Councillor McVey.
"One of the things that came out of the Edinburgh Poverty Commission recommendations was around the idea of "pram-push" distance - that if services were out of reach, beyond that pram-push distance, then immediately we were creating barriers to people being able to access what should be universal services, whether that be health, community, education services or access to employment."
The policy could mean some public services, such as libraries and council offices, being relocated to share premises - and that might include rationalisation.
But Councillor McVey says changes will be based on the idea of a genuine community. "It's not just 'we'll quarter the city and provide services on that level because that suits the council or NHS', but looking at communities as they are for the people who live there and then working out what services we need to provide there.
"It's also central to the work on carbon reduction because the more we can have people living, working, shopping, getting everything they need within their own locality they will travel less across the city which will reduce emissions."
The council wants the 20-minute neighbourhood idea to be integral to new developments in the city - and it is already happening.
The original plan for redevelopment of the former Powderhall Waste Transfer Station site was to put it on the market, where it was expected to be bought by a volume housebuilder. But instead the council is now taking forward a major mixed-use regeneration project and the proposed masterplan has been expanded to include adjacent disused bowling greens.
Instead of a "typical infill housing scheme", the project will deliver not only around 260 homes, at least 35 per cent of them affordable, but also a new 128-place nursery, seven new commercial units and new civic and green space. And the adjacent B-listed historic former stable block is to be refurbished into creative workspaces and community events space.
The stable block, which is phase one of the regeneration and due for completion within months, will be heated using air-source heat pumps as part of the project's drive for zero-carbon standards.
In Wester Hailes, a 10-15 year masterplan is being drawn up for a phased approach to a 20-minute neighbourhood.
In what the council calls "early action projects", work is already under way on improvements to existing council homes and estates, delivery of new-build affordable homes at Dumbryden Gardens, designing a replacement high school and improvements to Westside Plaza.
In the longer term, the council says there are opportunities to consider new projects, such as redeveloping the site of the existing Calder Glen Nursery and former Westburn Primary School for housing but with a replacement nursery twice the size of the current one and specialist housing for the elderly and vulnerable groups.
And a new community hub in Pennywell will provide the communities of North Edinburgh with much-needed childcare places, a new library and learning hub, and integrated cultural, arts and community support.
In Liberton, where a new high school is already planned, the council says the agreement to acquire the Liberton hospital site for affordable housing offers an ideal opportunity to co-ordinate the two projects through the 20-minute neighbourhood programme.
The high school could become a community campus, offering a range of services and activities at different times of the day and evening, including options for a GP service and a Police Scotland presence. Walking, cycling and public transport routes would include links to the new homes on the former hospital site.
And the council also wants to bring 20-minute neighbourhood principles to well-established communities.
Work is under way with community groups in Portobello, looking at various publicly-used buildings within walking distance of the town centre and opportunities for co-location of services.
The council says Portobello could become a local pilot and support the principle of working closer to home.
In Craigmillar, improved connections are being planned to the local high street in Niddrie Mains Road and a new multi-use community public space known as "Walk-Up Avenue" will offer a place for people to meet outdoors in the town centre, including seating, informal play and events space, a stage pavilion and a sculptural entranceway.
And in Corstorphine, improvements are being planned to St John's Road and the planned transfer of council-owned Westfield House to the community has, according to the council, created the opportunity to consider wider consolidation of community services, including the potential relocation of the existing library to the ground floor of the building.
But does the emphasis on local communities pose a threat to the future of a thriving city centre? Not according to Councillor McVey.
"One of the things that underpins our approach is access to those things which are not going to be co-located within communities. Some things in the city centre can't be replicated in every community: we need to have a secondary conversation about how do we connect in those communities to citywide assets, world class museums, culture, music – a lot of that is still in the city centre and essential for people's wellbeing.
"The 20-minute neighbourhood is not about confining people to their own 20 minute space, it's about making sure they have what they need on their doorstep when they need it, but for those things that are important to our experience within the city they still have easy access to that."
He also acknowledges it is unrealistic for everyone to work in their own neighbourhood.
"This is not about the entire economy of that area being centred within a 20-minute radius.
"Within every community in Edinburgh there will be people who work at the university or tech companies or finance, and when not working from home those people will be based at certain locations around the city.
"But for some people having to travel to work immediately becomes a barrier, maybe because of childcare, so it is really important that within every 20-minute neighbourhood we have local employment opportunities. There shouldn't be 20-minute neighbourhoods where employment is not present."
And the council has acknowledged the importance of involving residents in the decisions on changes in their area.
A recent report said: "Building trusted relationships with the people who live or work in neighbourhoods and therefore are best placed to know what is and is not good about it will be at the centre of any plans for 20-minute neighbourhood proposals. Communities helping shape proposals will be key to success."