David Halpern, in The Hidden Wealth of Nations, reported on the Blair administration's attempts to improve the lot of the worst off in Britain by removing barriers to achievement. To the discomfort of the left, they discovered that removing barriers didn't do much to reduce inequality because many people at the bottom didn't want to seize more opportunities (and more work).
He also reports that most nations are not opposed to inequality, if they think the process that produced it is basically fair. This is true regardless of how unequal that nation's economic condition actually is.
So Halpern suggests a sensible centrist aim for government. The state should focus on fostering decency, mutual respect, and access to basic services, especially on the part of the state itself. The government should not put its main effort into eliminating inequality or poverty.