It has been a good week for President Obama — apart from the moment when Colonel Muammar Gaddafi called him “my son”. He has won the support of Russia, and maybe China, on pressing Iran to curb its nuclear ambitions. The remarks by President Medvedev were conditional, to be sure, but solid enough that Obama could stand smiling by his side and announce his success to the world.

In opening the UN Security Council session yesterday he called for “the day when nuclear weapons have been banned”. He put this theme, toxically eccentric a few years ago, right at the top of his personal agenda.

His magical appeal to other leaders was intact. They clustered round him, Gordon Brown managing almost half a minute of face time before the superhero turned away, and a beaming David Miliband securing a handshake in the seconds before the session.

But there have been shadows. Afghanistan is a black hole — Obama barely mentioned it, and no wonder, given that he is in a quandary about whether to scrap the “surge” he announced just six months ago.

He has only a handful of loose sand to show on the Middle East, even though Tony Blair said yesterday that Israeli-Palestinian talks would restart within weeks. Obama did reassert, in front of the UN General Assembly, that Israel must halt expansion of West Bank settlements, trying to counter speculation that he had backed down in the face of Israeli intransigence. He said the US did Israel no favours by indulging it — but also that the UN did the Palestinians no favours by refusing to accept any responsibility for the solution.

But such careful symmetry may not translate into progress. This week’s briefings by George Mitchell, his special envoy, were revealing — and dispiriting — in showing his taste for oblique anecdote and exposition. They did not inspire confidence in America’s ability to drive hard for a result.

In Pittsburgh today, as Obama opens the G20 meeting of countries that make up 85 per cent of the world’s economic output, he can take comfort from the apparent success of his $800 billion stimulus and $700 billion bank bailout. They may yet prove his most solid triumphs. But his decision last week to slap tariffs on Chinese tyres will let poorer countries brand him a hypocrite in his call for free trade. It also leaves him on the back foot with China.

Most uncomfortably for Obama, while this week has reinforced foreign adoration, it has fired up with even more passion those Americans who detest him. The home front remains his toughest battleground: healthcare, then carbon emissions, then the economy again.

A President whose trademark is speeches of perfect balance, aiming to unite warring factions, is powerfully divisive within America. If he fails to overcome that, foreign policy offers a refuge. But, although many foreign leaders would continue to adore him, his influence abroad will dwindle if he cannot also succeed at home.

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