To win again on 2 November Mr Dinkins must hold on to his black voters, keep his Jewish liberal supporters and take his share of the fastest-growing bloc of Hispanic voters. His backing in all three groups is frayed at the edges.
Most blacks will vote for Mr Dinkins, no doubt. He won 91 per cent of the black vote in 1989 and that accounted for 28 per cent of the ballots cast. But many blacks are disappointed that the hopes and dreams associated with a black mayor have not been realised. Unemployment in New York is 9 per cent, or three points higher than the national average and blacks are disproportionately affected. Blacks also feel they have been ignored by the mayor in favour of Hispanics and Jews.
Enthusiasm among Hispanics, now a hugely expanded group from the Puerto Ricans, Cubans and Dominicans that existed 10 years ago, has waned. In the last contest, 64 per cent of the Hispanics voted for Mr Dinkins. His support has since dropped into the forties, with Mr Giuliani's climbing. Part of the reason is the newcomers' perception of the city as a place in the grip of progressive social deterioration. Entire families are wiped out in drugs wars and schoolchildren are killed for a pair of Mickey Mouse sunglasses. Mr Dinkins may have increased the number of police on the beat, but he has not raised the city's quality of life.
Jews started to turn against Mr Dinkins after the 1991 Crown Heights riots in Brooklyn in which an Hasidic scholar, Yankel Rosenbaum, was stabbed to death by a group of young blacks. The official report on the riots that came out in May blamed the mayor for not acting sooner to curb the violence, which lasted for three days.
The mayor's support among Jewish voters has fallen to 23 per cent from one-third in May, and to 21 per cent among white voters overall, down from 27 per cent in the 1989 election.
To woo the key liberal Jewish voter - the kind of life-long Democrats who used to think their right arm would fall off if they voted Republican - Mr Giuliani has been trying to soften his established image as a tough no-nonsense law-and-order Republican who favours bringing back the death penalty to New York state and locking up violent offenders without parole. After several notable Giuliani successes in cracking down on mobsters, President George Bush once called him 'America's greatest crime-fighter', in a league with the likes of Eliot Ness.
It is apparent that liberal thoughts do not come naturally to Mr Giuliani, but, like any leading American politician, he has his image-makers and they have had him walking into poor neighbourhoods and throwing baseballs to children to show his human side. His physical appearance does not help this transformation. His face is dour and angular, with deep-set eyes and thin lips.
Some of his policies also turn off liberals. In the only city in the US that guarantees every person a right to shelter, he proposes kicking the homeless back on to the streets after 90 days. That is the only way they will learn to pick themselves up and join society, he says. 'The less you expect of people, the less you get.' At the same time, he says: 'I love my wife and kids. I don't kick dogs. I'm a nice guy. This idea that I'm mean-hearted is a myth.'
Such a tight contest tends to bring out the worst in the candidates and their supporters. In one of the latest exchanges, black Baptist ministers supporting the mayor accused Mr Giuliani of having fascist tendencies, like Mussolini, Hitler and Franco. They said Mr Giuliani and his camp 'had no respect for the truth' in blaming Mr Dinkins for the Crown Heights riots. The implication was that Mr Giuliani was also a racist.
In other American political campaigns, candidates have apologised for the use by supporters of such harsh and patently ludicrous terms to describe their opponents, but not in the New York mayoral race. 'This is not my characterisation (of Mr Giuliani),' said the mayor, adding: 'I have always said we live in a racist society, that's not new.'
A month before the polls, it is not clear to cynical New Yorkers that either candidate can provide what they are looking for: a job, and not to be mugged. The latest poll indicates fewer than half of the voters trust either candidate to deal with the city's problems.