PARIS (AP) - One thing is certain in the race to lead France's cultural and political center: A woman will be mayor of Paris for the first time in the city's 2,000-year history.

The outcome of the conservative primary that begins on May 31 is all but decided - Nathalie Kosciusko-Morizet, or NKM as she is often known, is widely considered the only candidate with a realistic chance.

Her Socialist opponent in the March 2014 election will be Anne Hidalgo, the current mayor's designated heir.

The two have already begun to spar indirectly, notably over security and tourism in Paris, where ugly riots erupted earlier this month during a celebration to honour the French soccer club Paris Saint-Germain. But they have distinctly different visions of how Paris should serve its 2.3 million residents and the 29 million people who visit each year.

The race also includes other female candidates from smaller parties who are considered unlikely to win.

Ms Kosciusko-Morizet has called for stores in the city's main tourist districts to open on Sundays, saying that Paris is losing tour groups to London on the weekends because of requirements that shops close for a day. She also wants to crack down on the pickpockets who swarm the subways and major attractions such as the Louvre and Eiffel Tower.

"We have something to learn about hospitality," she said.

Ms Hidalgo counters that the French system works for its residents, saying that she does not want Paris - which virtually shuts down on Sundays and in the evenings - to "look like Anglo-Saxon cities working 24 hours a day."

Ms Kosciusko-Morizet, 40, is an engineer with deep family roots in France's political world - her grandfather was once ambassador to the United States and her father is mayor of a small town on the outskirts of the capital. She herself was mayor of the Paris suburb of Longjumeau until this year.

She also led the wide-ranging ministry of Ecology, Sustainable Development, Transport and Housing under former President Nicolas Sarkozy, where she was seen as a tech-savvy and ambitious star in the conservative UMP party. She was the spokesman for Mr Sarkozy's failed presidential re-election campaign last year and remains a deputy in the National Assembly.

In contrast, Ms Hidalgo, deputy to Mayor Bertrand Delanoe, who is retiring after 12 years in office, is more reticent in public. At a recent visit to a street market, she remained firmly surrounded by aides who handed out literature. She refused multiple requests for an interview with The Associated Press, and only rarely appears on television or comments in French newspapers.

But Ms Hidalgo, 53, whose parents emigrated from Spain when she was 2, could benefit from Mr Delanoe's popularity and Paris' system of indirect voting, in which the mayor assumes power upon the vote of leaders of individual neighborhoods.

She was holding the first public meeting of her campaign on Tuesday night in a trendy concert hall.

Just five of the 40 French towns with populations greater than 100,000 have female mayors, testament to French women's difficulty in getting top political jobs.

In 2000, France passed a law requiring gender parity among candidates, but the country still ranks low in global comparisons for women's political empowerment. The UN's 2012 Global Gender Gap survey placed France at 63 in the world, between Ethiopia and Chile.

"I'm happy there are a lot of women in this Parisian battle - it's proof that Paris has evolved as a city. At least, those criticisms we often hear about the credibility or competence of women won't be an issue," Ms Hidalgo said after one debate.

"It's true that there aren't enough women in French politics. And I'm an engineer, there are also few engineers in French politics," said Ms Kosciusko-Morizet.

In truth, the two women share many positions: Both back public housing, mindful that Paris is among the most expensive cities in the world. Both are calling for limits on diesel vehicles and both support gay marriage, which was recently legalised in France.

"This situation - where most of the candidates are women- is unprecedented in France," said Gael Sliman, a political analyst who also runs the polling agency BVA Opinion.

The candidates "are quite young and vibrant, with a modern style - completely the opposite of the usual profile of the old, grizzled, lifeless politician. They totally fit the Parisian voters."

Sandrine Leveque, an expert on gender issues at the Sorbonne University in Paris, agreed that Paris was the most likely French city to produce female mayoral candidates. Still, she predicted the campaign will take a toll on Ms Hidalgo and Ms Kosciusko-Morizet, saying French female politicians "are in an impossible situation."

"When they act too much like women, we say they act too much like women," Ms Leveque noted, "When they don't act enough like women, we say that they are not woman enough."