I want to get GeoJSON data for the UK. The regions which I require are in the map displayed in the below link: Former Government Office regions
I have been searching the net but all in vain. Can anyone help?
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Geographic data are rarely distributed as GeoJson files. If you need to do some search, you should rather enter "download shapefile" as key words. For the UK, you can find free administrative boundaries on the Ordnance survey website or on GDAM.
Once you have your shapefile, you can convert it to Geojson using different software. For instance, you can use Python (see here) or QGIS (Layer > save layer as... GeoJson).
I've also found an online converter but did not test it.