(Adds Putin quotes, latest medals)
* Austrian Mayer wins men's downhill
* Russia hoping for first gold in team figure skating
* Gay athlete says fuss over controversial law exaggerated
* Graf bags first medal for hosts
By Mike Collett-White
SOCHI, Russia, Feb 9 (Reuters) - Austrian Matthias Mayer conquered the treacherous Rosa Khutor run at Russia's Winter Olympics on Sunday to win the men's downhill, while the host nation was eyeing its first gold medal of the Games in the inaugural figure skating team event.
On the second day of full competition on Russia's Black Sea coast, Mayer claimed one of the Sochi Games' biggest titles, as pre-race favourites American Bode Miller and Norway's Aksel Lund Svindal failed to make the podium.
Son of 1988 super-G silver medallist Helmut, Mayer edged out Italy's Christof Innerhofer by 0.06 seconds on an overcast day in the Caucasus mountains.
"It's crazy. It's the greatest thing you can achieve as a sportsman - unbelievable," said the 23-year-old.
Russian President Vladimir Putin, who has staked his reputation on staging a successful Games, said hosting the event had helped shelter the country from economic crisis.
Allegations of widespread corruption have dogged the huge seven-year infrastructure project, and with a price tag of more than $50 billion, it is the most expensive Olympics ever.
"It is fully justified to say that the Olympic project, the Olympic construction work as a whole, was one of the most significant anti-crisis measures in the country," Putin said in comments to state TV broadcaster, Rossiya 24, aired on Sunday.
Putin says there is no evidence of major corruption in Sochi, but a recent survey by independent pollster Levada showed 47 percent of Russians believe the cost of the Games has soared because funds have been embezzled or mismanaged.
RUSSIA'S FIRST MEDAL
On the ice, Irene Wust of the Netherlands won the women's 3,000 metre Olympic crown, but the biggest roar went to Olga Graf who finished a surprise third to give the host nation its first medal.
Graf punched the air in delight upon realising her time but her expression turned to embarrassment soon after as she unzipped her suit to the waist before suddenly realising and grabbing the zip edges to protect her modesty.
"I heard the crowd cheering for me and I didn't expect such support from the audience," said the skater, before addressing her wardrobe malfunction.
"I totally forgot that I had nothing under my suit," she said, her steely focus disintegrating into a broad smile.
Putin congratulated his compatriot: "I am sure your success will boost the spirit of the Russian Olympians and give them added strength and confidence," he said.
Russia's best chance of gold on Sunday was in figure skating, after debutante Julia Lipnitskaya, 15, blew away more illustrious rivals with her short skate on Saturday and put the hosts within striking distance of the Olympic team title.
Swiss cross-country skier Dario Cologna surged to surprise victory in the men's skiathlon and Jamie Anderson clinched the inaugural women's snowboarding slopestyle gold after compatriot Sage Kotsenburg matched the feat in the men's event on Saturday.
Jenny Jones claimed the bronze -- Britain's first Olympic medal in a snow event.
Organisers said many ticket holders were not getting into venues because they were turning up too late, explaining some of the thousands of empty seats at skating and skiing events.
Even at the men's downhill there were several hundred vacant places in the stands at the bottom of the fearsome piste.
"The downhill is the king of the sport so I don't know what's going on. The Russians have no downhill skiers, that's the problem here," said Austria's 1980 Olympic champion Leonhard Stock, surveying stands that were more than half-empty only minutes before the scheduled start of the race.
Other venues have been far from full, and organisers urged ticket holders to turn up early because a sizeable number were not getting through security in time.
"A lot of people need to understand what the time for them to travel is (to the competitions)," Sochi Games spokeswoman Alexandra Kosterina told reporters.
"We are tying to alert people... to come in advance," she said, adding that the fact that around 10 percent of ticket-holders were missing their competitions was not necessarily down to the tight security at the Sochi Games.
She said 92 percent of tickets on offer on Saturday had been sold, while attendance was 81 percent.
THREATS OF ATTACK
Heightened security is largely explained by threats from Islamist militants in nearby Chechnya and neighbouring regions who have vowed to attack the Olympics, which they say take place on land seized from Caucasus tribes in the 19th century.
Some 37,000 security personnel are on high alert in and around Sochi, although officials believe the risk of militant violence is in fact greater elsewhere in Russia.
A major attack during the Games would embarrass Putin, who launched a war to crush a rebellion in Chechnya in 1999.
A gunman killed a nun and churchgoer and wounded six other people on Sunday in a Russian Orthodox cathedral on the island of Sakhalin, thousands of kilometres (miles) to the east, but there was no apparent link to Sochi.
As well as international unease over the risk of violence, preparations for Russia's first ever Winter Games have been overshadowed by the outcry over recent legislation banning the promotion of gay propaganda among minors.
Russia believes the law is needed to protect minors, but activists say it fuels anti-gay violence.
Lesbian Austrian ski jumper and gay rights advocate Daniela Iraschko-Stolz said public reaction to the law had been exaggerated. The 30-year-old married her lesbian partner last year, a rare case of an Austrian athlete coming out as gay.
Asked whether she was worried about the law, Iraschko-Stolz replied: "No, on the contrary, I think everything is being blown up bigger than it is. I had a very good welcome like every other athlete: there were absolutely no problems."
Russian police detained gay rights activists who tried to protest on Moscow's Red Square and in St Petersburg on Friday, shortly before the Games opened. (Additional reporting by Patrick Johnston and Pritha Sarkar in Sochi and Alan Baldwin, David Ljunggren, Mark Trevelyan and Nick Mulvenney in Rosa Khutor; Editing by Ossian Shine)