Tenants with one spare bedroom have had a payment reduction of 14% and those deemed to have two or more spare a reduction of 25%.
Lawyers for the 10 cases say that, unless the families move from their homes into smaller properties, they face building up rent arrears and being forced out any way.
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Journalist Steve Hewlett, who shared his experience of battling cancer on BBC Radio 4, has died of the disease at the age of 58.
A long-time diplomat, Churkin had been Russia's permanent representative to the UN since 2006.