Judgment is being handed down by Mr Justice Ouseley at London's High Court. Whatever the judge decides, appeals against his decisions are almost certain to be taken to the Court of Appeal.
Key accusations include that the Government has unlawfully failed to undertake a strategic environmental assessment of the entire project before giving the go-ahead, or to arrange a consultation process "that allows informed responses on the true merits and demerits of the proposals".
There is also a dispute over the compensation arrangements being put in place for homeowners living near the HS2 line.
Some 172,000 properties within 1km of the first phase are reported to be affected by "a generalised blight" since the proposals were announced.
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A picture of Cecil the lion - the animal killed by an American dentist in Zimbabwe - has been projected on to the Empire State Building.
A man, believed to be in his 50s, was pronounced dead at the scene after apparently suffering a heart attack.